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

<b>set </b><font color="Maroon" title="c1">V</font> =  <a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>();<br/>
<b>set </b><font color="Maroon" title="c2">X</font> = <span class="p1">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c3">A</font> =  <a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p1">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c4">U</font> = the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c5">D</font> =  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c6">G</font> =  <a href="msafree.html#K5" title="MSAFREE:func.5">DTConMSA</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <span class="p1">(<span class="default"><a href="msafree.html#K5" title="MSAFREE:func.5">DTConMSA</a> <span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span>) <b>-&gt; </b>  <a href="abcmiz_1.html#NM6" title="ABCMIZ_1:NM.6">expression</a> of <font color="Maroon">F<sub>1</sub></font>() = <font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default">$1 <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>);<br/>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <span class="p1">(<span class="default"><a href="msafree.html#K5" title="MSAFREE:func.5">DTConMSA</a> <span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span>,  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>,  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> =  <a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> <span class="p1">(<span class="default">$1 <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>,<a href="abcmiz_1.html#K9" title="ABCMIZ_1:func.9">*</a> ,<font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default">$3 <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>,<span class="p1">(<span class="default">$3 <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</span>)</span>),<span class="p1">(<span class="default"><a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> <span class="p2">(<span class="default">$1 <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>,<a href="abcmiz_1.html#K10" title="ABCMIZ_1:func.10">non_op</a> ,<font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default">$3 <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>),<font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default">$1 <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>,$3)</span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c7">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="msafree.html#K5" title="MSAFREE:func.5">DTConMSA</a> <span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span>, <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span><b> such that </b><br/><a NAME="E2:236_1"/><i><font color="Green" title="E120">A6</font></i>: 
for <font color="Olive" title="b1">t</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <span class="p1">(<span class="default"><a href="msafree.html#K5" title="MSAFREE:func.5">DTConMSA</a> <span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span>  st <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="abcmiz_1.html#K51" title="ABCMIZ_1:func.51">Terminals</a> <span class="p1">(<span class="default"><a href="msafree.html#K5" title="MSAFREE:func.5">DTConMSA</a> <span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> holds <br/><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Olive" title="b1">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">t</font>)
 <b>and </b><br/><a NAME="E3:236_1"/><i><font color="Green" title="E121">A7</font></i>: 
for <font color="Olive" title="b1">nt</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <span class="p1">(<span class="default"><a href="msafree.html#K5" title="MSAFREE:func.5">DTConMSA</a> <span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span><br/> for <font color="Olive" title="b2">ts</font> being   <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="msafree.html#K5" title="MSAFREE:func.5">DTConMSA</a> <span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span>  st <font color="Olive" title="b1">nt</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Olive" title="b2">ts</font> holds <br/><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Olive" title="b2">ts</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">nt</font>, <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Olive" title="b2">ts</font>,<font color="Maroon" title="c7">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b2">ts</font>)
 <b>from </b><i><a class="ref" href="dtconstr.html#S8" title="DTCONSTR:sch.8">DTCONSTR:sch 8</a>();<br/></i>
<a NAME="E4:236_1"/><i><font color="Green" title="E122">A5</font></i>: 
 <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="msafree.html#K5" title="MSAFREE:func.5">DTConMSA</a> <span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="abcmiz_1.html#T128" target="_self" title="ABCMIZ_1:th.128">LemTS</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">f</font> = <font color="Maroon" title="c7">f</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span>, <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> ;<br/>
<a NAME="E6:236_1"/>
<font color="Maroon" title="c8">f</font> is    <a href="abcmiz_1.html#M4" title="ABCMIZ_1:mode.4">term-transformation</a> of <font color="Maroon">F<sub>1</sub></font>(), <a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="236_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">s</font> be   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>(); <i><font color="Red">:: according to </font></i><a class="ref" href="abcmiz_1.html#D56" target="_self" title="ABCMIZ_1:def.56">ABCMIZ_1:def 56</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p2">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p3">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">s</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">s</font></span><br/><b>let </b><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p2">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p3">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">s</font></span>)</span> or <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">s</font> )</span><br/>



<b>assume </b><a NAME="E1:236_1_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p2">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p3">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">s</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">s</font></span><br/>

<b>then consider </b><font color="Maroon" title="c11">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:236_1_1"/><i><font color="Green" title="E123">B1</font></i>: 
( <font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">s</font> &amp; <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">a</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="abcmiz_1.html#NM6" title="ABCMIZ_1:NM.6">expression</a> of <font color="Maroon">F<sub>1</sub></font>()] <b>means </b>for <font color="Olive" title="b1">s</font> being  <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>()  st $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font> holds <br/><font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font>;<br/>
<a NAME="E4:236_1_1"/><i><font color="Green" title="E124">W1</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="abcmiz_1.html#NM3" title="ABCMIZ_1:NM.3">variable</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font> <a href="abcmiz_1.html#K35" title="ABCMIZ_1:func.35">-term</a> <font color="Maroon">F<sub>1</sub></font>()]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="236_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">y</font> be   <a href="abcmiz_1.html#NM3" title="ABCMIZ_1:NM.3">variable</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">y</font> <a href="abcmiz_1.html#K35" title="ABCMIZ_1:func.35">-term</a> <font color="Maroon">F<sub>1</sub></font>()]</span><br/>

<b>set </b><font color="Maroon" title="c13">a</font> = <font color="Maroon" title="c12">y</font> <a href="abcmiz_1.html#K35" title="ABCMIZ_1:func.35">-term</a> <font color="Maroon">F<sub>1</sub></font>();<br/>
<b>let </b><font color="Maroon" title="c14">s</font> be   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>(); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c12">y</font> <a href="abcmiz_1.html#K35" title="ABCMIZ_1:func.35">-term</a> <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">s</font> implies <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">y</font> <a href="abcmiz_1.html#K35" title="ABCMIZ_1:func.35">-term</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">s</font> )</span><br/>

<b>assume </b><a NAME="E1:236_1_1_1"/><i><font color="Green" title="E125">C1</font></i>: 
<font color="Maroon" title="c12">y</font> <a href="abcmiz_1.html#K35" title="ABCMIZ_1:func.35">-term</a> <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">y</font> <a href="abcmiz_1.html#K35" title="ABCMIZ_1:func.35">-term</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">s</font></span><br/>

<a NAME="E2:236_1_1_1"/><i><font color="Green" title="E126">C2</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">y</font>,<span class="p2">(<span class="default"><a href="abcmiz_1.html#K14" title="ABCMIZ_1:func.14">a_Term</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="abcmiz_1.html#K51" title="ABCMIZ_1:func.51">Terminals</a> <span class="p1">(<span class="default"><a href="msafree.html#K5" title="MSAFREE:func.5">DTConMSA</a> <span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="abcmiz_1.html#T122" target="_self" title="ABCMIZ_1:th.122">LemTerm</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c15">t</font> = <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">y</font>,<span class="p2">(<span class="default"><a href="abcmiz_1.html#K14" title="ABCMIZ_1:func.14">a_Term</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> as   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <span class="p1">(<span class="default"><a href="msafree.html#K5" title="MSAFREE:func.5">DTConMSA</a> <span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> ;<br/>
<font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">y</font> <a href="abcmiz_1.html#K35" title="ABCMIZ_1:func.35">-term</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> = 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c15">t</font>)
<b>by </b><i><a class="txt" href="abcmiz_1.html#E2:236_1"><i><font color="Green" title="E120">A6</font></i></a>, <a class="txt" href="abcmiz_1.html#E2:236_1_1_1"><i><font color="Green" title="E126">C2</font></i></a></i>
<br/>.= 
<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c12">y</font>)
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
;<br/>
<a NAME="E5:236_1_1_1"/><b>then </b><i><font color="Green" title="E127">C3</font></i>: 
( <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">y</font> <a href="abcmiz_1.html#K35" title="ABCMIZ_1:func.35">-term</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> is   <a href="abcmiz_1.html#NM9" title="ABCMIZ_1:NM.9">quasi-term</a> of <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon" title="c12">y</font> <a href="abcmiz_1.html#K35" title="ABCMIZ_1:func.35">-term</a> <font color="Maroon">F<sub>1</sub></font>() is    <a href="abcmiz_1.html#M1" title="ABCMIZ_1:mode.1">expression</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon" title="c14">s</font> )
 
<b>by </b><i><a class="txt" href="abcmiz_1.html#E1:236"><i><font color="Green" title="E116">A1</font></i></a>, <a class="txt" href="abcmiz_1.html#E1:236_1_1_1"><i><font color="Green" title="E125">C1</font></i></a>, <a class="ref" href="abcmiz_1.html#D28" target="_self" title="ABCMIZ_1:def.28">ELEMENT</a></i>;<br/>
<a NAME="E6:236_1_1_1"/><b>then </b>
<font color="Maroon" title="c14">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="abcmiz_1.html#K14" title="ABCMIZ_1:func.14">a_Term</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="ref" href="abcmiz_1.html#T48" target="_self" title="ABCMIZ_1:th.48">Th90A</a></i>;<br/>
<b>hence </b><a NAME="E7:236_1_1_1"/>
<font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">y</font> <a href="abcmiz_1.html#K35" title="ABCMIZ_1:func.35">-term</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">s</font>
 <b>by </b><i><a class="txt" href="abcmiz_1.html#E5:236_1_1_1"><i><font color="Green" title="E127">C3</font></i></a>, <a class="ref" href="abcmiz_1.html#D28" target="_self" title="ABCMIZ_1:def.28">ELEMENT</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E5:236_1_1"/><i><font color="Green" title="E125">W2</font></i>: 
for <font color="Olive" title="b1">c</font> being <a href="abcmiz_1.html#V2" title="ABCMIZ_1:attr.2">constructor</a> <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive" title="b2">p</font> being   <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="abcmiz_1.html#K34" title="ABCMIZ_1:func.34">QuasiTerms</a> <font color="Maroon">F<sub>1</sub></font>()  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Olive" title="b1">c</font></span>)</span> &amp; ( for <font color="Olive" title="b3">t</font> being  <a href="abcmiz_1.html#NM9" title="ABCMIZ_1:NM.9">quasi-term</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive" title="b3">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b2">p</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b3">t</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">c</font> <a href="abcmiz_1.html#K36" title="ABCMIZ_1:func.36">-trm</a> <font color="Olive" title="b2">p</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="236_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">c</font> be  <a href="abcmiz_1.html#V2" title="ABCMIZ_1:attr.2">constructor</a> <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon">F<sub>1</sub></font>(); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">p</font> being   <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="abcmiz_1.html#K34" title="ABCMIZ_1:func.34">QuasiTerms</a> <font color="Maroon">F<sub>1</sub></font>()  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c12">c</font></span>)</span> &amp; ( for <font color="Olive" title="b2">t</font> being  <a href="abcmiz_1.html#NM9" title="ABCMIZ_1:NM.9">quasi-term</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive" title="b2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">p</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">t</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">c</font> <a href="abcmiz_1.html#K36" title="ABCMIZ_1:func.36">-trm</a> <font color="Olive" title="b1">p</font>]</span><br/><b>let </b><font color="Maroon" title="c13">p</font> be    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="abcmiz_1.html#K34" title="ABCMIZ_1:func.34">QuasiTerms</a> <font color="Maroon">F<sub>1</sub></font>(); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c12">c</font></span>)</span> &amp; ( for <font color="Olive" title="b1">t</font> being  <a href="abcmiz_1.html#NM9" title="ABCMIZ_1:NM.9">quasi-term</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c13">p</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">t</font>] ) implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">c</font> <a href="abcmiz_1.html#K36" title="ABCMIZ_1:func.36">-trm</a> <font color="Maroon" title="c13">p</font>] )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:236_1_1_2"/><i><font color="Green" title="E126">Z0</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c12">c</font></span>)</span>
 <b>and </b><br/><a NAME="E2:236_1_1_2"/><i><font color="Green" title="E127">Z1</font></i>: 
for <font color="Olive" title="b1">t</font> being  <a href="abcmiz_1.html#NM9" title="ABCMIZ_1:NM.9">quasi-term</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c13">p</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">t</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">c</font> <a href="abcmiz_1.html#K36" title="ABCMIZ_1:func.36">-trm</a> <font color="Maroon" title="c13">p</font>]</span><br/>

<b>set </b><font color="Maroon" title="c14">a</font> = <font color="Maroon" title="c12">c</font> <a href="abcmiz_1.html#K36" title="ABCMIZ_1:func.36">-trm</a> <font color="Maroon" title="c13">p</font>;<br/>
<b>set </b><font color="Maroon" title="c15">nt</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">c</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>
<b>let </b><font color="Maroon" title="c16">s</font> be   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>(); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c12">c</font> <a href="abcmiz_1.html#K36" title="ABCMIZ_1:func.36">-trm</a> <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">s</font> implies <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">c</font> <a href="abcmiz_1.html#K36" title="ABCMIZ_1:func.36">-trm</a> <font color="Maroon" title="c13">p</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">s</font> )</span><br/>

<b>assume </b><a NAME="E3:236_1_1_2"/><i><font color="Green" title="E128">00</font></i>: 
<font color="Maroon" title="c12">c</font> <a href="abcmiz_1.html#K36" title="ABCMIZ_1:func.36">-trm</a> <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">c</font> <a href="abcmiz_1.html#K36" title="ABCMIZ_1:func.36">-trm</a> <font color="Maroon" title="c13">p</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">s</font></span><br/>

<a NAME="E4:236_1_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">c</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></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="abcmiz_1.html#K52" title="ABCMIZ_1:func.52">NonTerminals</a> <span class="p1">(<span class="default"><a href="msafree.html#K5" title="MSAFREE:func.5">DTConMSA</a> <span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="abcmiz_1.html#T123" target="_self" title="ABCMIZ_1:th.123">LemNTerm</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c17">nt</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">c</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <span class="p1">(<span class="default"><a href="msafree.html#K5" title="MSAFREE:func.5">DTConMSA</a> <span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c18">ts</font> = <font color="Maroon" title="c13">p</font> as    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="msafree.html#K5" title="MSAFREE:func.5">DTConMSA</a> <span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <b>by </b><i><a class="txt" href="abcmiz_1.html#E4:236_1"><i><font color="Green" title="E122">A5</font></i></a></i>;<br/>
<a NAME="E7:236_1_1_2"/><i><font color="Green" title="E129">Z3</font></i>: 
<font color="Maroon" title="c12">c</font> <a href="abcmiz_1.html#K36" title="ABCMIZ_1:func.36">-trm</a> <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c18">ts</font>
 
<b>by </b><i><a class="txt" href="abcmiz_1.html#E1:236_1_1_2"><i><font color="Green" title="E126">Z0</font></i></a>, <a class="ref" href="abcmiz_1.html#D35" target="_self" title="ABCMIZ_1:def.35">TERM</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c19">aa</font> = <font color="Maroon" title="c12">c</font> <a href="abcmiz_1.html#K36" title="ABCMIZ_1:func.36">-trm</a> <font color="Maroon" title="c13">p</font> as   <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon">F<sub>1</sub></font>(),<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="msafree3.html#T9" title="MSAFREE3:th.9">MSAFREE3:9</a></i>;<br/>
<a NAME="E9:236_1_1_2"/>
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon">F<sub>1</sub></font>() <a href="msafree3.html#K5" title="MSAFREE3:func.5">-Terms</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="msafree3.html#T25" title="MSAFREE3:th.25">MSAFREE3:25</a></i>;<br/>
<a NAME="E10:236_1_1_2"/><b>then </b>
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p3">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E11:236_1_1_2"/><b>then </b>
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p3">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">s</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a></i>;<br/>
<a NAME="E12:236_1_1_2"/><b>then </b>
<font color="Maroon" title="c19">aa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="msafree.html#K8" title="MSAFREE:func.8">FreeSort</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p3">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">s</font>
 
<b>by </b><i><a class="txt" href="abcmiz_1.html#E3:236_1_1_2"><i><font color="Green" title="E128">00</font></i></a></i>;<br/>
<a NAME="E13:236_1_1_2"/><b>then </b>
<font color="Maroon" title="c19">aa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree.html#K7" title="MSAFREE:func.7">FreeSort</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>,<font color="Maroon" title="c16">s</font>
 
<b>by </b><i><a class="ref" href="msafree.html#D13" title="MSAFREE:def.13">MSAFREE:def 13</a></i>;<br/>
<a NAME="E14:236_1_1_2"/><b>then </b><i><font color="Green" title="E130">01</font></i>: 
 <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c19">aa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">s</font>
 
<b>by </b><i><a class="ref" href="msaterm.html#D5" title="MSATERM:def.5">MSATERM:def 5</a></i>;<br/>
<a NAME="E15:236_1_1_2"/><i><font color="Green" title="E131">Z4</font></i>: 
( <font color="Maroon" title="c17">nt</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">c</font> &amp; <font color="Maroon" title="c12">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="abcmiz_1.html#K9" title="ABCMIZ_1:func.9">*</a>  &amp; <font color="Maroon" title="c12">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="abcmiz_1.html#K10" title="ABCMIZ_1:func.10">non_op</a>  )
 
<b>by </b><i><a class="ref" href="abcmiz_1.html#D11" target="_self" title="ABCMIZ_1:def.11">CNSTR2</a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>
<a NAME="E16:236_1_1_2"/><i><font color="Green" title="E132">Z5</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c13">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="abcmiz_1.html#K34" title="ABCMIZ_1:func.34">QuasiTerms</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E17:236_1_1_2"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E18:236_1_1_2"/><b>then </b><i><font color="Green" title="E133">Z7</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c13">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">f</font>
 
;<br/>
<a NAME="E19:236_1_1_2"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c13">p</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="abcmiz_1.html#K34" title="ABCMIZ_1:func.34">QuasiTerms</a> <font color="Maroon">F<sub>1</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="236_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c20">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c20">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c13">p</font></span>)</span> or <font color="Maroon" title="c20">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="abcmiz_1.html#K34" title="ABCMIZ_1:func.34">QuasiTerms</a> <font color="Maroon">F<sub>1</sub></font>() )</span><br/>

<b>assume </b><a NAME="E1:236_1_1_2_1"/>
<font color="Maroon" title="c20">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c13">p</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c20">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="abcmiz_1.html#K34" title="ABCMIZ_1:func.34">QuasiTerms</a> <font color="Maroon">F<sub>1</sub></font>()</span><br/>

<b>then consider </b><font color="Maroon" title="c21">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:236_1_1_2_1"/><i><font color="Green" title="E134">V1</font></i>: 
( <font color="Maroon" title="c21">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c13">p</font></span>)</span> &amp; <font color="Maroon" title="c20">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c13">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:236_1_1_2_1"/>
<font color="Maroon" title="c21">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">p</font>
 
<b>by </b><i><a class="txt" href="abcmiz_1.html#E3:236_1_1_2_1"><i><font color="Green" title="E134">V1</font></i></a>, <a class="txt" href="abcmiz_1.html#E18:236_1_1_2"><i><font color="Green" title="E133">Z7</font></i></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/>
<a NAME="E5:236_1_1_2_1"/><b>then </b><i><font color="Green" title="E135">V2</font></i>: 
<font color="Maroon" title="c13">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c13">p</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c22">pi</font> = <font color="Maroon" title="c13">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> as   <a href="abcmiz_1.html#NM9" title="ABCMIZ_1:NM.9">quasi-term</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="txt" href="abcmiz_1.html#E16:236_1_1_2"><i><font color="Green" title="E132">Z5</font></i></a>, <a class="ref" href="abcmiz_1.html#T41" target="_self" title="ABCMIZ_1:th.41">Th42</a></i>;<br/>
<a NAME="E7:236_1_1_2_1"/>
( <font color="Maroon" title="c22">pi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K14" title="ABCMIZ_1:func.14">a_Term</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c22">pi</font>] )
 
<b>by </b><i><a class="txt" href="abcmiz_1.html#E2:236_1_1_2"><i><font color="Green" title="E127">Z1</font></i></a>, <a class="txt" href="abcmiz_1.html#E5:236_1_1_2_1"><i><font color="Green" title="E135">V2</font></i></a>, <a class="ref" href="abcmiz_1.html#T41" target="_self" title="ABCMIZ_1:th.41">Th42</a></i>;<br/>
<a NAME="E8:236_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">pi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K14" title="ABCMIZ_1:func.14">a_Term</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>
 
;<br/>
<b>hence </b><a NAME="E9:236_1_1_2_1"/>
<font color="Maroon" title="c20">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="abcmiz_1.html#K34" title="ABCMIZ_1:func.34">QuasiTerms</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>by </b><i><a class="txt" href="abcmiz_1.html#E3:236_1_1_2_1"><i><font color="Green" title="E134">V1</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c20">q</font> = <font color="Maroon" title="c8">f</font> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c13">p</font> as    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="abcmiz_1.html#K34" title="ABCMIZ_1:func.34">QuasiTerms</a> <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E21:236_1_1_2"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c13">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>1</sub></font>() <a href="msaterm.html#K1" title="MSATERM:func.1">-Terms</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="236_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c21">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c21">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c13">p</font> or <font color="Maroon" title="c21">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() <a href="msaterm.html#K1" title="MSATERM:func.1">-Terms</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:236_1_1_2_2"/>
<font color="Maroon" title="c21">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c13">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c21">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() <a href="msaterm.html#K1" title="MSATERM:func.1">-Terms</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span><br/>

<a NAME="E2:236_1_1_2_2"/><b>then </b>
<font color="Maroon" title="c21">z</font> is    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="msaterm.html#K1" title="MSATERM:func.1">-Terms</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="msafree3.html#T9" title="MSAFREE3:th.9">MSAFREE3:9</a></i>;<br/>
<b>hence </b><a NAME="E3:236_1_1_2_2"/>
<font color="Maroon" title="c21">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() <a href="msaterm.html#K1" title="MSATERM:func.1">-Terms</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c21">r</font> = <font color="Maroon" title="c13">p</font> as    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="msaterm.html#K1" title="MSATERM:func.1">-Terms</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E23:236_1_1_2"/><i><font color="Green" title="E134">Z6</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">p</font>
 
<b>by </b><i><a class="txt" href="abcmiz_1.html#E18:236_1_1_2"><i><font color="Green" title="E133">Z7</font></i></a>, <a class="ref" href="finseq_2.html#T33" title="FINSEQ_2:th.33">FINSEQ_2:33</a></i>;<br/>
<a NAME="E24:236_1_1_2"/>
<font color="Maroon" title="c12">c</font> <a href="abcmiz_1.html#K36" title="ABCMIZ_1:func.36">-trm</a> <font color="Maroon" title="c13">p</font> is   <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon">F<sub>1</sub></font>(),<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="msafree3.html#T9" title="MSAFREE3:th.9">MSAFREE3:9</a></i>;<br/>
<a NAME="E25:236_1_1_2"/><b>then </b>
<font color="Maroon" title="c21">r</font> is    <a href="msaterm.html#M1" title="MSATERM:mode.1">ArgumentSeq</a> of  <a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> <font color="Maroon" title="c12">c</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="abcmiz_1.html#E7:236_1_1_2"><i><font color="Green" title="E129">Z3</font></i></a>, <a class="ref" href="msaterm.html#T1" title="MSATERM:th.1">MSATERM:1</a></i>;<br/>
<a NAME="E26:236_1_1_2"/><b>then </b><i><font color="Green" title="E135">02</font></i>: 
(  <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c12">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">s</font> &amp;  <a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> <font color="Maroon" title="c12">c</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c21">r</font> )
 
<b>by </b><i><a class="txt" href="abcmiz_1.html#E14:236_1_1_2"><i><font color="Green" title="E130">01</font></i></a>, <a class="txt" href="abcmiz_1.html#E7:236_1_1_2"><i><font color="Green" title="E129">Z3</font></i></a>, <a class="ref" href="msaterm.html#T20" title="MSATERM:th.20">MSATERM:20</a>, <a class="ref" href="msaterm.html#T21" title="MSATERM:th.21">MSATERM:21</a></i>;<br/>
<a NAME="E27:236_1_1_2"/><b>then </b>
<font color="Maroon" title="c17">nt</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c18">ts</font>
 
<b>by </b><i><a class="ref" href="abcmiz_1.html#T127" target="_self" title="ABCMIZ_1:th.127">LemArr</a></i>;<br/>
<b>then </b><font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">c</font> <a href="abcmiz_1.html#K36" title="ABCMIZ_1:func.36">-trm</a> <font color="Maroon" title="c13">p</font></span>)</span> = 
<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c17">nt</font>, <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c18">ts</font>,<font color="Maroon" title="c8">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c18">ts</font>)
<b>by </b><i><a class="txt" href="abcmiz_1.html#E3:236_1"><i><font color="Green" title="E121">A7</font></i></a>, <a class="txt" href="abcmiz_1.html#E7:236_1_1_2"><i><font color="Green" title="E129">Z3</font></i></a></i>
<br/>.= 
 <a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> <font color="Maroon" title="c12">c</font>,<a href="abcmiz_1.html#K10" title="ABCMIZ_1:func.10">non_op</a> ,<font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c18">ts</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>),<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c12">c</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c18">ts</font></span>)</span>)
<b>by </b><i><a class="txt" href="abcmiz_1.html#E15:236_1_1_2"><i><font color="Green" title="E131">Z4</font></i></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>
<br/>.= 
<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c12">c</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c18">ts</font></span>)</span>)
<b>by </b><i><a class="txt" href="abcmiz_1.html#E15:236_1_1_2"><i><font color="Green" title="E131">Z4</font></i></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>
;<br/>
<a NAME="E29:236_1_1_2"/><b>then </b>
<font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">c</font> <a href="abcmiz_1.html#K36" title="ABCMIZ_1:func.36">-trm</a> <font color="Maroon" title="c13">p</font></span>)</span> is    <a href="abcmiz_1.html#M1" title="ABCMIZ_1:mode.1">expression</a> of <font color="Maroon">F<sub>1</sub></font>(), <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c12">c</font>
 
<b>by </b><i><a class="txt" href="abcmiz_1.html#E1:236_1_1_2"><i><font color="Green" title="E126">Z0</font></i></a>, <a class="txt" href="abcmiz_1.html#E23:236_1_1_2"><i><font color="Green" title="E134">Z6</font></i></a>, <a class="txt" href="abcmiz_1.html#E2:236"><i><font color="Green" title="E117">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E30:236_1_1_2"/>
<font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">c</font> <a href="abcmiz_1.html#K36" title="ABCMIZ_1:func.36">-trm</a> <font color="Maroon" title="c13">p</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">s</font>
 <b>by </b><i><a class="txt" href="abcmiz_1.html#E26:236_1_1_2"><i><font color="Green" title="E135">02</font></i></a>, <a class="ref" href="abcmiz_1.html#D28" target="_self" title="ABCMIZ_1:def.28">ELEMENT</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E6:236_1_1"/><i><font color="Green" title="E126">W3</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="abcmiz_1.html#M1" title="ABCMIZ_1:mode.1">expression</a> of <font color="Maroon">F<sub>1</sub></font>(), <a href="abcmiz_1.html#K13" title="ABCMIZ_1:func.13">an_Adj</a> <font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<span class="p1">(<span class="default"><a href="abcmiz_1.html#K32" title="ABCMIZ_1:func.32">non_op</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="abcmiz_1.html#K30" title="ABCMIZ_1:func.30">term</a> <font color="Olive" title="b1">a</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="236_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">v</font> be    <a href="abcmiz_1.html#M1" title="ABCMIZ_1:mode.1">expression</a> of <font color="Maroon">F<sub>1</sub></font>(), <a href="abcmiz_1.html#K13" title="ABCMIZ_1:func.13">an_Adj</a> <font color="Maroon">F<sub>1</sub></font>(); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">v</font>] implies <font color="Maroon">S<sub>1</sub></font>[<span class="p1">(<span class="default"><a href="abcmiz_1.html#K32" title="ABCMIZ_1:func.32">non_op</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="abcmiz_1.html#K30" title="ABCMIZ_1:func.30">term</a> <font color="Maroon" title="c12">v</font>] )</span><br/>

<b>assume </b><a NAME="E1:236_1_1_3"/><i><font color="Green" title="E127">Z1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">v</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<span class="p1">(<span class="default"><a href="abcmiz_1.html#K32" title="ABCMIZ_1:func.32">non_op</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="abcmiz_1.html#K30" title="ABCMIZ_1:func.30">term</a> <font color="Maroon" title="c12">v</font>]</span><br/>

<a NAME="E2:236_1_1_3"/><i><font color="Green" title="E128">Y1</font></i>: 
<font color="Maroon" title="c12">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K13" title="ABCMIZ_1:func.13">an_Adj</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>
 
<b>by </b><i><a class="ref" href="abcmiz_1.html#D28" target="_self" title="ABCMIZ_1:def.28">ELEMENT</a></i>;<br/>
<a NAME="E3:236_1_1_3"/><b>then </b><i><font color="Green" title="E129">Y2</font></i>: 
<font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K13" title="ABCMIZ_1:func.13">an_Adj</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>
 
<b>by </b><i><a class="txt" href="abcmiz_1.html#E1:236_1_1_3"><i><font color="Green" title="E127">Z1</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c13">fv</font> = <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">v</font> as    <a href="abcmiz_1.html#M1" title="ABCMIZ_1:mode.1">expression</a> of <font color="Maroon">F<sub>1</sub></font>(), <a href="abcmiz_1.html#K13" title="ABCMIZ_1:func.13">an_Adj</a> <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="ref" href="abcmiz_1.html#D28" target="_self" title="ABCMIZ_1:def.28">ELEMENT</a></i>;<br/>
<b>let </b><font color="Maroon" title="c14">s</font> be   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>(); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="abcmiz_1.html#K32" title="ABCMIZ_1:func.32">non_op</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="abcmiz_1.html#K30" title="ABCMIZ_1:func.30">term</a> <font color="Maroon" title="c12">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">s</font> implies <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K32" title="ABCMIZ_1:func.32">non_op</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="abcmiz_1.html#K30" title="ABCMIZ_1:func.30">term</a> <font color="Maroon" title="c12">v</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">s</font> )</span><br/>

<b>assume </b><a NAME="E5:236_1_1_3"/><i><font color="Green" title="E130">00</font></i>: 
<span class="p1">(<span class="default"><a href="abcmiz_1.html#K32" title="ABCMIZ_1:func.32">non_op</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="abcmiz_1.html#K30" title="ABCMIZ_1:func.30">term</a> <font color="Maroon" title="c12">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K32" title="ABCMIZ_1:func.32">non_op</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="abcmiz_1.html#K30" title="ABCMIZ_1:func.30">term</a> <font color="Maroon" title="c12">v</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">s</font></span><br/>

<a NAME="E6:236_1_1_3"/><b>then </b>
( <span class="p1">(<span class="default"><a href="abcmiz_1.html#K32" title="ABCMIZ_1:func.32">non_op</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="abcmiz_1.html#K30" title="ABCMIZ_1:func.30">term</a> <font color="Maroon" title="c12">v</font> is    <a href="abcmiz_1.html#M1" title="ABCMIZ_1:mode.1">expression</a> of <font color="Maroon">F<sub>1</sub></font>(), <a href="abcmiz_1.html#K13" title="ABCMIZ_1:func.13">an_Adj</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <span class="p1">(<span class="default"><a href="abcmiz_1.html#K32" title="ABCMIZ_1:func.32">non_op</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="abcmiz_1.html#K30" title="ABCMIZ_1:func.30">term</a> <font color="Maroon" title="c12">v</font> is    <a href="abcmiz_1.html#M1" title="ABCMIZ_1:mode.1">expression</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon" title="c14">s</font> )
 
<b>by </b><i><a class="ref" href="abcmiz_1.html#D28" target="_self" title="ABCMIZ_1:def.28">ELEMENT</a>, <a class="ref" href="abcmiz_1.html#T43" target="_self" title="ABCMIZ_1:th.43">ThNon</a></i>;<br/>
<a NAME="E7:236_1_1_3"/><b>then </b><i><font color="Green" title="E131">03</font></i>: 
<font color="Maroon" title="c14">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="abcmiz_1.html#K13" title="ABCMIZ_1:func.13">an_Adj</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="ref" href="abcmiz_1.html#T48" target="_self" title="ABCMIZ_1:th.48">Th90A</a></i>;<br/>
<b>set </b><font color="Maroon" title="c15">QA</font> = the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K13" title="ABCMIZ_1:func.13">an_Adj</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>;<br/>
<a NAME="E8:236_1_1_3"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c12">v</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c12">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T55" title="FINSEQ_1:th.55">FINSEQ_1:55</a></i>;<br/>
<a NAME="E9:236_1_1_3"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c12">v</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K13" title="ABCMIZ_1:func.13">an_Adj</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>
 
<b>by </b><i><a class="txt" href="abcmiz_1.html#E2:236_1_1_3"><i><font color="Green" title="E128">Y1</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c16">p</font> = <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c12">v</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K13" title="ABCMIZ_1:func.13">an_Adj</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<b>set </b><font color="Maroon" title="c17">c</font> =  <a href="abcmiz_1.html#K32" title="ABCMIZ_1:func.32">non_op</a> <font color="Maroon">F<sub>1</sub></font>();<br/>
<b>set </b><font color="Maroon" title="c18">a</font> = <span class="p1">(<span class="default"><a href="abcmiz_1.html#K32" title="ABCMIZ_1:func.32">non_op</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="abcmiz_1.html#K30" title="ABCMIZ_1:func.30">term</a> <font color="Maroon" title="c12">v</font>;<br/>
<b>set </b><font color="Maroon" title="c19">nt</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K32" title="ABCMIZ_1:func.32">non_op</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>
<a NAME="E11:236_1_1_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K32" title="ABCMIZ_1:func.32">non_op</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></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="abcmiz_1.html#K52" title="ABCMIZ_1:func.52">NonTerminals</a> <span class="p1">(<span class="default"><a href="msafree.html#K5" title="MSAFREE:func.5">DTConMSA</a> <span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="abcmiz_1.html#T123" target="_self" title="ABCMIZ_1:th.123">LemNTerm</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c20">nt</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K32" title="ABCMIZ_1:func.32">non_op</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <span class="p1">(<span class="default"><a href="msafree.html#K5" title="MSAFREE:func.5">DTConMSA</a> <span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c21">ts</font> = <font color="Maroon" title="c16">p</font> as    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="msafree.html#K5" title="MSAFREE:func.5">DTConMSA</a> <span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <b>by </b><i><a class="txt" href="abcmiz_1.html#E4:236_1"><i><font color="Green" title="E122">A5</font></i></a></i>;<br/>
<a NAME="E14:236_1_1_3"/><i><font color="Green" title="E132">Z3</font></i>: 
<span class="p1">(<span class="default"><a href="abcmiz_1.html#K32" title="ABCMIZ_1:func.32">non_op</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="abcmiz_1.html#K30" title="ABCMIZ_1:func.30">term</a> <font color="Maroon" title="c12">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c21">ts</font>
 
<b>by </b><i><a class="ref" href="abcmiz_1.html#T43" target="_self" title="ABCMIZ_1:th.43">ThNon</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c22">aa</font> = <span class="p1">(<span class="default"><a href="abcmiz_1.html#K32" title="ABCMIZ_1:func.32">non_op</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="abcmiz_1.html#K30" title="ABCMIZ_1:func.30">term</a> <font color="Maroon" title="c12">v</font> as   <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon">F<sub>1</sub></font>(),<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="msafree3.html#T9" title="MSAFREE3:th.9">MSAFREE3:9</a></i>;<br/>
<a NAME="E16:236_1_1_3"/>
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon">F<sub>1</sub></font>() <a href="msafree3.html#K5" title="MSAFREE3:func.5">-Terms</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="msafree3.html#T25" title="MSAFREE3:th.25">MSAFREE3:25</a></i>;<br/>
<a NAME="E17:236_1_1_3"/><b>then </b>
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p3">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E18:236_1_1_3"/><b>then </b>
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p3">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">s</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a></i>;<br/>
<a NAME="E19:236_1_1_3"/><b>then </b>
<font color="Maroon" title="c22">aa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="msafree.html#K8" title="MSAFREE:func.8">FreeSort</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p3">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">s</font>
 
<b>by </b><i><a class="txt" href="abcmiz_1.html#E5:236_1_1_3"><i><font color="Green" title="E130">00</font></i></a></i>;<br/>
<a NAME="E20:236_1_1_3"/><b>then </b>
<font color="Maroon" title="c22">aa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree.html#K7" title="MSAFREE:func.7">FreeSort</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>,<font color="Maroon" title="c14">s</font>
 
<b>by </b><i><a class="ref" href="msafree.html#D13" title="MSAFREE:def.13">MSAFREE:def 13</a></i>;<br/>
<a NAME="E21:236_1_1_3"/><b>then </b><i><font color="Green" title="E133">01</font></i>: 
 <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c22">aa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">s</font>
 
<b>by </b><i><a class="ref" href="msaterm.html#D5" title="MSATERM:def.5">MSATERM:def 5</a></i>;<br/>
<a NAME="E22:236_1_1_3"/><i><font color="Green" title="E134">Z4</font></i>: 
( <font color="Maroon" title="c20">nt</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="abcmiz_1.html#K32" title="ABCMIZ_1:func.32">non_op</a> <font color="Maroon">F<sub>1</sub></font>() &amp;  <a href="abcmiz_1.html#K32" title="ABCMIZ_1:func.32">non_op</a> <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="abcmiz_1.html#K9" title="ABCMIZ_1:func.9">*</a>  )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>
<a NAME="E23:236_1_1_3"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E24:236_1_1_3"/><b>then </b><i><font color="Green" title="E135">08</font></i>: 
<font color="Maroon" title="c8">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c16">p</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c13">fv</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="bagorder.html#T3" title="BAGORDER:th.3">BAGORDER:3</a></i>;<br/>
<a NAME="E25:236_1_1_3"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c13">fv</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c13">fv</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T55" title="FINSEQ_1:th.55">FINSEQ_1:55</a></i>;<br/>
<a NAME="E26:236_1_1_3"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c13">fv</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K13" title="ABCMIZ_1:func.13">an_Adj</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>
 
<b>by </b><i><a class="txt" href="abcmiz_1.html#E3:236_1_1_3"><i><font color="Green" title="E129">Y2</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c23">q</font> = <font color="Maroon" title="c8">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c16">p</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K13" title="ABCMIZ_1:func.13">an_Adj</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <b>by </b><i><a class="txt" href="abcmiz_1.html#E24:236_1_1_3"><i><font color="Green" title="E135">08</font></i></a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E28:236_1_1_3"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c16">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>1</sub></font>() <a href="msaterm.html#K1" title="MSATERM:func.1">-Terms</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="236_1_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c24">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c24">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c16">p</font> or <font color="Maroon" title="c24">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() <a href="msaterm.html#K1" title="MSATERM:func.1">-Terms</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:236_1_1_3_1"/>
<font color="Maroon" title="c24">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c16">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c24">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() <a href="msaterm.html#K1" title="MSATERM:func.1">-Terms</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span><br/>

<a NAME="E2:236_1_1_3_1"/><b>then </b>
<font color="Maroon" title="c24">z</font> is    <a href="abcmiz_1.html#M1" title="ABCMIZ_1:mode.1">expression</a> of <font color="Maroon">F<sub>1</sub></font>(), <a href="abcmiz_1.html#K13" title="ABCMIZ_1:func.13">an_Adj</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="ref" href="abcmiz_1.html#T41" target="_self" title="ABCMIZ_1:th.41">Th42</a></i>;<br/>
<a NAME="E3:236_1_1_3_1"/><b>then </b>
<font color="Maroon" title="c24">z</font> is    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="msaterm.html#K1" title="MSATERM:func.1">-Terms</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="msafree3.html#T9" title="MSAFREE3:th.9">MSAFREE3:9</a></i>;<br/>
<b>hence </b><a NAME="E4:236_1_1_3_1"/>
<font color="Maroon" title="c24">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() <a href="msaterm.html#K1" title="MSATERM:func.1">-Terms</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c24">r</font> = <font color="Maroon" title="c16">p</font> as    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="msaterm.html#K1" title="MSATERM:func.1">-Terms</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E30:236_1_1_3"/>
<span class="p1">(<span class="default"><a href="abcmiz_1.html#K32" title="ABCMIZ_1:func.32">non_op</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="abcmiz_1.html#K30" title="ABCMIZ_1:func.30">term</a> <font color="Maroon" title="c12">v</font> is   <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon">F<sub>1</sub></font>(),<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="msafree3.html#T9" title="MSAFREE3:th.9">MSAFREE3:9</a></i>;<br/>
<a NAME="E31:236_1_1_3"/><b>then </b>
<font color="Maroon" title="c24">r</font> is    <a href="msaterm.html#M1" title="MSATERM:mode.1">ArgumentSeq</a> of  <a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K32" title="ABCMIZ_1:func.32">non_op</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="abcmiz_1.html#E14:236_1_1_3"><i><font color="Green" title="E132">Z3</font></i></a>, <a class="ref" href="msaterm.html#T1" title="MSATERM:th.1">MSATERM:1</a></i>;<br/>
<a NAME="E32:236_1_1_3"/><b>then </b>
(  <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K32" title="ABCMIZ_1:func.32">non_op</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">s</font> &amp;  <a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K32" title="ABCMIZ_1:func.32">non_op</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c24">r</font> )
 
<b>by </b><i><a class="txt" href="abcmiz_1.html#E21:236_1_1_3"><i><font color="Green" title="E133">01</font></i></a>, <a class="txt" href="abcmiz_1.html#E14:236_1_1_3"><i><font color="Green" title="E132">Z3</font></i></a>, <a class="ref" href="msaterm.html#T20" title="MSATERM:th.20">MSATERM:20</a>, <a class="ref" href="msaterm.html#T21" title="MSATERM:th.21">MSATERM:21</a></i>;<br/>
<a NAME="E33:236_1_1_3"/><b>then </b>
<font color="Maroon" title="c20">nt</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c21">ts</font>
 
<b>by </b><i><a class="ref" href="abcmiz_1.html#T127" target="_self" title="ABCMIZ_1:th.127">LemArr</a></i>;<br/>
<b>then </b><font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K32" title="ABCMIZ_1:func.32">non_op</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="abcmiz_1.html#K30" title="ABCMIZ_1:func.30">term</a> <font color="Maroon" title="c12">v</font></span>)</span> = 
<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c20">nt</font>, <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c21">ts</font>,<font color="Maroon" title="c8">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c21">ts</font>)
<b>by </b><i><a class="txt" href="abcmiz_1.html#E3:236_1"><i><font color="Green" title="E121">A7</font></i></a>, <a class="txt" href="abcmiz_1.html#E14:236_1_1_3"><i><font color="Green" title="E132">Z3</font></i></a></i>
<br/>.= 
 <a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K32" title="ABCMIZ_1:func.32">non_op</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,<a href="abcmiz_1.html#K10" title="ABCMIZ_1:func.10">non_op</a> ,<font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c21">ts</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>),<font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><a href="abcmiz_1.html#K32" title="ABCMIZ_1:func.32">non_op</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c21">ts</font></span>)</span>)
<b>by </b><i><a class="txt" href="abcmiz_1.html#E22:236_1_1_3"><i><font color="Green" title="E134">Z4</font></i></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>
<br/>.= 
<font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c21">ts</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>)
<b>by </b><i><a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>
<br/>.= 
<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c13">fv</font>)
<b>by </b><i><a class="txt" href="abcmiz_1.html#E24:236_1_1_3"><i><font color="Green" title="E135">08</font></i></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>
;<br/>
<a NAME="E35:236_1_1_3"/><b>then </b>
<font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K32" title="ABCMIZ_1:func.32">non_op</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="abcmiz_1.html#K30" title="ABCMIZ_1:func.30">term</a> <font color="Maroon" title="c12">v</font></span>)</span> is    <a href="abcmiz_1.html#M1" title="ABCMIZ_1:mode.1">expression</a> of <font color="Maroon">F<sub>1</sub></font>(), <a href="abcmiz_1.html#K13" title="ABCMIZ_1:func.13">an_Adj</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="txt" href="abcmiz_1.html#E3:236"><i><font color="Green" title="E118">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E36:236_1_1_3"/>
<font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K32" title="ABCMIZ_1:func.32">non_op</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="abcmiz_1.html#K30" title="ABCMIZ_1:func.30">term</a> <font color="Maroon" title="c12">v</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">s</font>
 <b>by </b><i><a class="txt" href="abcmiz_1.html#E7:236_1_1_3"><i><font color="Green" title="E131">03</font></i></a>, <a class="ref" href="abcmiz_1.html#D28" target="_self" title="ABCMIZ_1:def.28">ELEMENT</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E7:236_1_1"/><i><font color="Green" title="E127">W4</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="abcmiz_1.html#M1" title="ABCMIZ_1:mode.1">expression</a> of <font color="Maroon">F<sub>1</sub></font>(), <a href="abcmiz_1.html#K13" title="ABCMIZ_1:func.13">an_Adj</a> <font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>] holds <br/>for <font color="Olive" title="b2">t</font> being   <a href="abcmiz_1.html#M1" title="ABCMIZ_1:mode.1">expression</a> of <font color="Maroon">F<sub>1</sub></font>(), <a href="abcmiz_1.html#K12" title="ABCMIZ_1:func.12">a_Type</a> <font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">t</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<span class="p1">(<span class="default"><a href="abcmiz_1.html#K33" title="ABCMIZ_1:func.33">ast</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="abcmiz_1.html#K31" title="ABCMIZ_1:func.31">term</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">t</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="236_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">v</font> be    <a href="abcmiz_1.html#M1" title="ABCMIZ_1:mode.1">expression</a> of <font color="Maroon">F<sub>1</sub></font>(), <a href="abcmiz_1.html#K13" title="ABCMIZ_1:func.13">an_Adj</a> <font color="Maroon">F<sub>1</sub></font>(); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">v</font>] implies for <font color="Olive" title="b1">t</font> being   <a href="abcmiz_1.html#M1" title="ABCMIZ_1:mode.1">expression</a> of <font color="Maroon">F<sub>1</sub></font>(), <a href="abcmiz_1.html#K12" title="ABCMIZ_1:func.12">a_Type</a> <font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">t</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<span class="p1">(<span class="default"><a href="abcmiz_1.html#K33" title="ABCMIZ_1:func.33">ast</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="abcmiz_1.html#K31" title="ABCMIZ_1:func.31">term</a> <font color="Maroon" title="c12">v</font>,<font color="Olive" title="b1">t</font>] )</span><br/>

<b>assume </b><a NAME="E1:236_1_1_4"/><i><font color="Green" title="E128">Z0</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">v</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">t</font> being   <a href="abcmiz_1.html#M1" title="ABCMIZ_1:mode.1">expression</a> of <font color="Maroon">F<sub>1</sub></font>(), <a href="abcmiz_1.html#K12" title="ABCMIZ_1:func.12">a_Type</a> <font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">t</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<span class="p1">(<span class="default"><a href="abcmiz_1.html#K33" title="ABCMIZ_1:func.33">ast</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="abcmiz_1.html#K31" title="ABCMIZ_1:func.31">term</a> <font color="Maroon" title="c12">v</font>,<font color="Olive" title="b1">t</font>]</span><br/>

<b>let </b><font color="Maroon" title="c13">t</font> be    <a href="abcmiz_1.html#M1" title="ABCMIZ_1:mode.1">expression</a> of <font color="Maroon">F<sub>1</sub></font>(), <a href="abcmiz_1.html#K12" title="ABCMIZ_1:func.12">a_Type</a> <font color="Maroon">F<sub>1</sub></font>(); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c13">t</font>] implies <font color="Maroon">S<sub>1</sub></font>[<span class="p1">(<span class="default"><a href="abcmiz_1.html#K33" title="ABCMIZ_1:func.33">ast</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="abcmiz_1.html#K31" title="ABCMIZ_1:func.31">term</a> <font color="Maroon" title="c12">v</font>,<font color="Maroon" title="c13">t</font>] )</span><br/>

<b>assume </b><a NAME="E2:236_1_1_4"/><i><font color="Green" title="E129">Z1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c13">t</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<span class="p1">(<span class="default"><a href="abcmiz_1.html#K33" title="ABCMIZ_1:func.33">ast</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="abcmiz_1.html#K31" title="ABCMIZ_1:func.31">term</a> <font color="Maroon" title="c12">v</font>,<font color="Maroon" title="c13">t</font>]</span><br/>

<a NAME="E3:236_1_1_4"/>
( <font color="Maroon" title="c12">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K13" title="ABCMIZ_1:func.13">an_Adj</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> &amp; <font color="Maroon" title="c13">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K12" title="ABCMIZ_1:func.12">a_Type</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> )
 
<b>by </b><i><a class="ref" href="abcmiz_1.html#D28" target="_self" title="ABCMIZ_1:def.28">ELEMENT</a></i>;<br/>
<a NAME="E4:236_1_1_4"/><b>then </b><i><font color="Green" title="E130">Y2</font></i>: 
( <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K13" title="ABCMIZ_1:func.13">an_Adj</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> &amp; <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K12" title="ABCMIZ_1:func.12">a_Type</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> )
 
<b>by </b><i><a class="txt" href="abcmiz_1.html#E1:236_1_1_4"><i><font color="Green" title="E128">Z0</font></i></a>, <a class="txt" href="abcmiz_1.html#E2:236_1_1_4"><i><font color="Green" title="E129">Z1</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c14">fv</font> = <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">v</font> as    <a href="abcmiz_1.html#M1" title="ABCMIZ_1:mode.1">expression</a> of <font color="Maroon">F<sub>1</sub></font>(), <a href="abcmiz_1.html#K13" title="ABCMIZ_1:func.13">an_Adj</a> <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="ref" href="abcmiz_1.html#D28" target="_self" title="ABCMIZ_1:def.28">ELEMENT</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">ft</font> = <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">t</font> as    <a href="abcmiz_1.html#M1" title="ABCMIZ_1:mode.1">expression</a> of <font color="Maroon">F<sub>1</sub></font>(), <a href="abcmiz_1.html#K12" title="ABCMIZ_1:func.12">a_Type</a> <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="txt" href="abcmiz_1.html#E4:236_1_1_4"><i><font color="Green" title="E130">Y2</font></i></a>, <a class="ref" href="abcmiz_1.html#D28" target="_self" title="ABCMIZ_1:def.28">ELEMENT</a></i>;<br/>
<b>let </b><font color="Maroon" title="c16">s</font> be   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>(); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="abcmiz_1.html#K33" title="ABCMIZ_1:func.33">ast</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="abcmiz_1.html#K31" title="ABCMIZ_1:func.31">term</a> <font color="Maroon" title="c12">v</font>,<font color="Maroon" title="c13">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">s</font> implies <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K33" title="ABCMIZ_1:func.33">ast</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="abcmiz_1.html#K31" title="ABCMIZ_1:func.31">term</a> <font color="Maroon" title="c12">v</font>,<font color="Maroon" title="c13">t</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">s</font> )</span><br/>

<b>assume </b><a NAME="E7:236_1_1_4"/><i><font color="Green" title="E131">00</font></i>: 
<span class="p1">(<span class="default"><a href="abcmiz_1.html#K33" title="ABCMIZ_1:func.33">ast</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="abcmiz_1.html#K31" title="ABCMIZ_1:func.31">term</a> <font color="Maroon" title="c12">v</font>,<font color="Maroon" title="c13">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K33" title="ABCMIZ_1:func.33">ast</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="abcmiz_1.html#K31" title="ABCMIZ_1:func.31">term</a> <font color="Maroon" title="c12">v</font>,<font color="Maroon" title="c13">t</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">s</font></span><br/>

<a NAME="E8:236_1_1_4"/><b>then </b>
( <span class="p1">(<span class="default"><a href="abcmiz_1.html#K33" title="ABCMIZ_1:func.33">ast</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="abcmiz_1.html#K31" title="ABCMIZ_1:func.31">term</a> <font color="Maroon" title="c12">v</font>,<font color="Maroon" title="c13">t</font> is    <a href="abcmiz_1.html#M1" title="ABCMIZ_1:mode.1">expression</a> of <font color="Maroon">F<sub>1</sub></font>(), <a href="abcmiz_1.html#K12" title="ABCMIZ_1:func.12">a_Type</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <span class="p1">(<span class="default"><a href="abcmiz_1.html#K33" title="ABCMIZ_1:func.33">ast</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="abcmiz_1.html#K31" title="ABCMIZ_1:func.31">term</a> <font color="Maroon" title="c12">v</font>,<font color="Maroon" title="c13">t</font> is    <a href="abcmiz_1.html#M1" title="ABCMIZ_1:mode.1">expression</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon" title="c16">s</font> )
 
<b>by </b><i><a class="ref" href="abcmiz_1.html#D28" target="_self" title="ABCMIZ_1:def.28">ELEMENT</a>, <a class="ref" href="abcmiz_1.html#T46" target="_self" title="ABCMIZ_1:th.46">ThAst</a></i>;<br/>
<a NAME="E9:236_1_1_4"/><b>then </b><i><font color="Green" title="E132">03</font></i>: 
<font color="Maroon" title="c16">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="abcmiz_1.html#K12" title="ABCMIZ_1:func.12">a_Type</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="ref" href="abcmiz_1.html#T48" target="_self" title="ABCMIZ_1:th.48">Th90A</a></i>;<br/>
<b>set </b><font color="Maroon" title="c17">QA</font> = the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K12" title="ABCMIZ_1:func.12">a_Type</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">p</font> = <span class="p1"><a href="lang1.html#K4" title="LANG1:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c12">v</font>,<font color="Maroon" title="c13">t</font></span><a href="lang1.html#K4" title="LANG1:func.4">*&gt;</a></span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> ;<br/>
<b>set </b><font color="Maroon" title="c19">c</font> =  <a href="abcmiz_1.html#K33" title="ABCMIZ_1:func.33">ast</a> <font color="Maroon">F<sub>1</sub></font>();<br/>
<b>set </b><font color="Maroon" title="c20">a</font> = <span class="p1">(<span class="default"><a href="abcmiz_1.html#K33" title="ABCMIZ_1:func.33">ast</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="abcmiz_1.html#K31" title="ABCMIZ_1:func.31">term</a> <font color="Maroon" title="c12">v</font>,<font color="Maroon" title="c13">t</font>;<br/>
<b>set </b><font color="Maroon" title="c21">nt</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K33" title="ABCMIZ_1:func.33">ast</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>
<a NAME="E11:236_1_1_4"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K33" title="ABCMIZ_1:func.33">ast</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></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="abcmiz_1.html#K52" title="ABCMIZ_1:func.52">NonTerminals</a> <span class="p1">(<span class="default"><a href="msafree.html#K5" title="MSAFREE:func.5">DTConMSA</a> <span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="abcmiz_1.html#T123" target="_self" title="ABCMIZ_1:th.123">LemNTerm</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c22">nt</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K33" title="ABCMIZ_1:func.33">ast</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <span class="p1">(<span class="default"><a href="msafree.html#K5" title="MSAFREE:func.5">DTConMSA</a> <span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c23">ts</font> = <font color="Maroon" title="c18">p</font> as    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="msafree.html#K5" title="MSAFREE:func.5">DTConMSA</a> <span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <b>by </b><i><a class="txt" href="abcmiz_1.html#E4:236_1"><i><font color="Green" title="E122">A5</font></i></a></i>;<br/>
<a NAME="E14:236_1_1_4"/><i><font color="Green" title="E133">Z3</font></i>: 
<span class="p1">(<span class="default"><a href="abcmiz_1.html#K33" title="ABCMIZ_1:func.33">ast</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="abcmiz_1.html#K31" title="ABCMIZ_1:func.31">term</a> <font color="Maroon" title="c12">v</font>,<font color="Maroon" title="c13">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c23">ts</font>
 
<b>by </b><i><a class="ref" href="abcmiz_1.html#T46" target="_self" title="ABCMIZ_1:th.46">ThAst</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c24">aa</font> = <span class="p1">(<span class="default"><a href="abcmiz_1.html#K33" title="ABCMIZ_1:func.33">ast</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="abcmiz_1.html#K31" title="ABCMIZ_1:func.31">term</a> <font color="Maroon" title="c12">v</font>,<font color="Maroon" title="c13">t</font> as   <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon">F<sub>1</sub></font>(),<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="msafree3.html#T9" title="MSAFREE3:th.9">MSAFREE3:9</a></i>;<br/>
<a NAME="E16:236_1_1_4"/>
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon">F<sub>1</sub></font>() <a href="msafree3.html#K5" title="MSAFREE3:func.5">-Terms</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="msafree3.html#T25" title="MSAFREE3:th.25">MSAFREE3:25</a></i>;<br/>
<a NAME="E17:236_1_1_4"/><b>then </b>
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p3">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E18:236_1_1_4"/><b>then </b>
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p3">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">s</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a></i>;<br/>
<a NAME="E19:236_1_1_4"/><b>then </b>
<font color="Maroon" title="c24">aa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="msafree.html#K8" title="MSAFREE:func.8">FreeSort</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p3">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">s</font>
 
<b>by </b><i><a class="txt" href="abcmiz_1.html#E7:236_1_1_4"><i><font color="Green" title="E131">00</font></i></a></i>;<br/>
<a NAME="E20:236_1_1_4"/><b>then </b>
<font color="Maroon" title="c24">aa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree.html#K7" title="MSAFREE:func.7">FreeSort</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>,<font color="Maroon" title="c16">s</font>
 
<b>by </b><i><a class="ref" href="msafree.html#D13" title="MSAFREE:def.13">MSAFREE:def 13</a></i>;<br/>
<a NAME="E21:236_1_1_4"/><b>then </b><i><font color="Green" title="E134">01</font></i>: 
 <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c24">aa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">s</font>
 
<b>by </b><i><a class="ref" href="msaterm.html#D5" title="MSATERM:def.5">MSATERM:def 5</a></i>;<br/>
<a NAME="E22:236_1_1_4"/><i><font color="Green" title="E135">Z4</font></i>: 
( <font color="Maroon" title="c22">nt</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="abcmiz_1.html#K33" title="ABCMIZ_1:func.33">ast</a> <font color="Maroon">F<sub>1</sub></font>() &amp;  <a href="abcmiz_1.html#K33" title="ABCMIZ_1:func.33">ast</a> <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="abcmiz_1.html#K10" title="ABCMIZ_1:func.10">non_op</a>  )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>
<a NAME="E23:236_1_1_4"/><i><font color="Green" title="E136">08</font></i>: 
<font color="Maroon" title="c8">f</font> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c18">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="lang1.html#K4" title="LANG1:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c14">fv</font>,<font color="Maroon" title="c15">ft</font></span><a href="lang1.html#K4" title="LANG1:func.4">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T40" title="FINSEQ_2:th.40">FINSEQ_2:40</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c25">q</font> = <font color="Maroon" title="c8">f</font> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c18">p</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> ;<br/>
<a NAME="E25:236_1_1_4"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c18">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>1</sub></font>() <a href="msaterm.html#K1" title="MSATERM:func.1">-Terms</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="236_1_1_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c26">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c26">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c18">p</font> or <font color="Maroon" title="c26">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() <a href="msaterm.html#K1" title="MSATERM:func.1">-Terms</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:236_1_1_4_1"/>
<font color="Maroon" title="c26">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c18">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c26">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() <a href="msaterm.html#K1" title="MSATERM:func.1">-Terms</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span><br/>

<a NAME="E2:236_1_1_4_1"/><b>then </b>
<font color="Maroon" title="c26">z</font> is    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="msaterm.html#K1" title="MSATERM:func.1">-Terms</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="msafree3.html#T9" title="MSAFREE3:th.9">MSAFREE3:9</a></i>;<br/>
<b>hence </b><a NAME="E3:236_1_1_4_1"/>
<font color="Maroon" title="c26">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() <a href="msaterm.html#K1" title="MSATERM:func.1">-Terms</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c26">r</font> = <font color="Maroon" title="c18">p</font> as    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="msaterm.html#K1" title="MSATERM:func.1">-Terms</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E27:236_1_1_4"/>
<span class="p1">(<span class="default"><a href="abcmiz_1.html#K33" title="ABCMIZ_1:func.33">ast</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="abcmiz_1.html#K31" title="ABCMIZ_1:func.31">term</a> <font color="Maroon" title="c12">v</font>,<font color="Maroon" title="c13">t</font> is   <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon">F<sub>1</sub></font>(),<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="msafree3.html#T9" title="MSAFREE3:th.9">MSAFREE3:9</a></i>;<br/>
<a NAME="E28:236_1_1_4"/><b>then </b>
<font color="Maroon" title="c26">r</font> is    <a href="msaterm.html#M1" title="MSATERM:mode.1">ArgumentSeq</a> of  <a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K33" title="ABCMIZ_1:func.33">ast</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="abcmiz_1.html#E14:236_1_1_4"><i><font color="Green" title="E133">Z3</font></i></a>, <a class="ref" href="msaterm.html#T1" title="MSATERM:th.1">MSATERM:1</a></i>;<br/>
<a NAME="E29:236_1_1_4"/><b>then </b>
(  <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K33" title="ABCMIZ_1:func.33">ast</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">s</font> &amp;  <a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K33" title="ABCMIZ_1:func.33">ast</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c26">r</font> )
 
<b>by </b><i><a class="txt" href="abcmiz_1.html#E21:236_1_1_4"><i><font color="Green" title="E134">01</font></i></a>, <a class="txt" href="abcmiz_1.html#E14:236_1_1_4"><i><font color="Green" title="E133">Z3</font></i></a>, <a class="ref" href="msaterm.html#T20" title="MSATERM:th.20">MSATERM:20</a>, <a class="ref" href="msaterm.html#T21" title="MSATERM:th.21">MSATERM:21</a></i>;<br/>
<a NAME="E30:236_1_1_4"/><b>then </b>
<font color="Maroon" title="c22">nt</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c23">ts</font>
 
<b>by </b><i><a class="ref" href="abcmiz_1.html#T127" target="_self" title="ABCMIZ_1:th.127">LemArr</a></i>;<br/>
<b>then </b><font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K33" title="ABCMIZ_1:func.33">ast</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="abcmiz_1.html#K31" title="ABCMIZ_1:func.31">term</a> <font color="Maroon" title="c12">v</font>,<font color="Maroon" title="c13">t</font></span>)</span> = 
<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c22">nt</font>, <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c23">ts</font>,<font color="Maroon" title="c8">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c23">ts</font>)
<b>by </b><i><a class="txt" href="abcmiz_1.html#E3:236_1"><i><font color="Green" title="E121">A7</font></i></a>, <a class="txt" href="abcmiz_1.html#E14:236_1_1_4"><i><font color="Green" title="E133">Z3</font></i></a></i>
<br/>.= 
<font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c23">ts</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c23">ts</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</span>)</span>)
<b>by </b><i><a class="txt" href="abcmiz_1.html#E22:236_1_1_4"><i><font color="Green" title="E135">Z4</font></i></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>
<br/>.= 
<font color="Maroon">F<sub>5</sub></font>(<font color="Maroon" title="c14">fv</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c23">ts</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</span>)</span>)
<b>by </b><i><a class="txt" href="abcmiz_1.html#E23:236_1_1_4"><i><font color="Green" title="E136">08</font></i></a>, <a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>
<br/>.= 
<font color="Maroon">F<sub>5</sub></font>(<font color="Maroon" title="c14">fv</font>,<font color="Maroon" title="c15">ft</font>)
<b>by </b><i><a class="txt" href="abcmiz_1.html#E23:236_1_1_4"><i><font color="Green" title="E136">08</font></i></a>, <a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>
;<br/>
<a NAME="E32:236_1_1_4"/><b>then </b>
<font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K33" title="ABCMIZ_1:func.33">ast</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="abcmiz_1.html#K31" title="ABCMIZ_1:func.31">term</a> <font color="Maroon" title="c12">v</font>,<font color="Maroon" title="c13">t</font></span>)</span> is    <a href="abcmiz_1.html#M1" title="ABCMIZ_1:mode.1">expression</a> of <font color="Maroon">F<sub>1</sub></font>(), <a href="abcmiz_1.html#K12" title="ABCMIZ_1:func.12">a_Type</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="txt" href="abcmiz_1.html#E4:236"><i><font color="Green" title="E119">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E33:236_1_1_4"/>
<font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K33" title="ABCMIZ_1:func.33">ast</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="abcmiz_1.html#K31" title="ABCMIZ_1:func.31">term</a> <font color="Maroon" title="c12">v</font>,<font color="Maroon" title="c13">t</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">s</font>
 <b>by </b><i><a class="txt" href="abcmiz_1.html#E9:236_1_1_4"><i><font color="Green" title="E132">03</font></i></a>, <a class="ref" href="abcmiz_1.html#D28" target="_self" title="ABCMIZ_1:def.28">ELEMENT</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E8:236_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">a</font>]
 
<b>from </b><i><a class="ref" href="abcmiz_1.html#S2" target="_self" title="ABCMIZ_1:sch.2">ABCMIZ_1:sch 2</a>(<a class="txt" href="abcmiz_1.html#E4:236_1_1"><i><font color="Green" title="E124">W1</font></i></a>, <a class="txt" href="abcmiz_1.html#E5:236_1_1"><i><font color="Green" title="E125">W2</font></i></a>, <a class="txt" href="abcmiz_1.html#E6:236_1_1"><i><font color="Green" title="E126">W3</font></i></a>, <a class="txt" href="abcmiz_1.html#E7:236_1_1"><i><font color="Green" title="E127">W4</font></i></a>);<br/></i>
<b>hence </b><a NAME="E9:236_1_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">s</font>
 <b>by </b><i><a class="txt" href="abcmiz_1.html#E3:236_1_1"><i><font color="Green" title="E123">B1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c9">f</font> = <font color="Maroon" title="c8">f</font> as    <a href="abcmiz_1.html#M4" title="ABCMIZ_1:mode.4">term-transformation</a> of <font color="Maroon">F<sub>1</sub></font>(), <a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>() ;<br/>
<b>take </b>
<font color="Maroon" title="c9">f</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">x</font> being  <a href="abcmiz_1.html#NM3" title="ABCMIZ_1:NM.3">variable</a> holds  <font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="abcmiz_1.html#K35" title="ABCMIZ_1:func.35">-term</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">x</font>) ) &amp; ( for <font color="Olive" title="b1">c</font> being <a href="abcmiz_1.html#V2" title="ABCMIZ_1:attr.2">constructor</a> <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive" title="b2">p</font>, <font color="Olive" title="b3">q</font> being   <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="abcmiz_1.html#K34" title="ABCMIZ_1:func.34">QuasiTerms</a> <font color="Maroon">F<sub>1</sub></font>()  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Olive" title="b1">c</font></span>)</span> &amp; <font color="Olive" title="b3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Olive" title="b2">p</font> holds <br/><font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">c</font> <a href="abcmiz_1.html#K36" title="ABCMIZ_1:func.36">-trm</a> <font color="Olive" title="b2">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">c</font>,<font color="Olive" title="b3">q</font>) ) &amp; ( for <font color="Olive" title="b1">a</font> being   <a href="abcmiz_1.html#M1" title="ABCMIZ_1:mode.1">expression</a> of <font color="Maroon">F<sub>1</sub></font>(), <a href="abcmiz_1.html#K13" title="ABCMIZ_1:func.13">an_Adj</a> <font color="Maroon">F<sub>1</sub></font>() holds  <font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K32" title="ABCMIZ_1:func.32">non_op</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="abcmiz_1.html#K30" title="ABCMIZ_1:func.30">term</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font></span>)</span>) ) &amp; ( for <font color="Olive" title="b1">a</font> being   <a href="abcmiz_1.html#M1" title="ABCMIZ_1:mode.1">expression</a> of <font color="Maroon">F<sub>1</sub></font>(), <a href="abcmiz_1.html#K13" title="ABCMIZ_1:func.13">an_Adj</a> <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive" title="b2">t</font> being   <a href="abcmiz_1.html#M1" title="ABCMIZ_1:mode.1">expression</a> of <font color="Maroon">F<sub>1</sub></font>(), <a href="abcmiz_1.html#K12" title="ABCMIZ_1:func.12">a_Type</a> <font color="Maroon">F<sub>1</sub></font>() holds  <font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K33" title="ABCMIZ_1:func.33">ast</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="abcmiz_1.html#K31" title="ABCMIZ_1:func.31">term</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">t</font></span>)</span>) ) )</span><br/>

<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">c</font> being <a href="abcmiz_1.html#V2" title="ABCMIZ_1:attr.2">constructor</a> <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive" title="b2">p</font>, <font color="Olive" title="b3">q</font> being   <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="abcmiz_1.html#K34" title="ABCMIZ_1:func.34">QuasiTerms</a> <font color="Maroon">F<sub>1</sub></font>()  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Olive" title="b1">c</font></span>)</span> &amp; <font color="Olive" title="b3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Olive" title="b2">p</font> holds <br/><font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">c</font> <a href="abcmiz_1.html#K36" title="ABCMIZ_1:func.36">-trm</a> <font color="Olive" title="b2">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">c</font>,<font color="Olive" title="b3">q</font>) ) &amp; ( for <font color="Olive" title="b1">a</font> being   <a href="abcmiz_1.html#M1" title="ABCMIZ_1:mode.1">expression</a> of <font color="Maroon">F<sub>1</sub></font>(), <a href="abcmiz_1.html#K13" title="ABCMIZ_1:func.13">an_Adj</a> <font color="Maroon">F<sub>1</sub></font>() holds  <font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K32" title="ABCMIZ_1:func.32">non_op</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="abcmiz_1.html#K30" title="ABCMIZ_1:func.30">term</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font></span>)</span>) ) &amp; ( for <font color="Olive" title="b1">a</font> being   <a href="abcmiz_1.html#M1" title="ABCMIZ_1:mode.1">expression</a> of <font color="Maroon">F<sub>1</sub></font>(), <a href="abcmiz_1.html#K13" title="ABCMIZ_1:func.13">an_Adj</a> <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive" title="b2">t</font> being   <a href="abcmiz_1.html#M1" title="ABCMIZ_1:mode.1">expression</a> of <font color="Maroon">F<sub>1</sub></font>(), <a href="abcmiz_1.html#K12" title="ABCMIZ_1:func.12">a_Type</a> <font color="Maroon">F<sub>1</sub></font>() holds  <font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K33" title="ABCMIZ_1:func.33">ast</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="abcmiz_1.html#K31" title="ABCMIZ_1:func.31">term</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">t</font></span>)</span>) ) )</span>
<div class="add"><b>let </b><font color="Maroon" title="c10">x</font> be   <a href="abcmiz_1.html#NM3" title="ABCMIZ_1:NM.3">variable</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="abcmiz_1.html#K35" title="ABCMIZ_1:func.35">-term</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c10">x</font>)</span><br/><a NAME="E1:236_1_2"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="abcmiz_1.html#K2" title="ABCMIZ_1:func.2">Vars</a> 
 ;<br/><a NAME="E2:236_1_2"/><b>then </b><i><font color="Green" title="E123">00</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K14" title="ABCMIZ_1:func.14">a_Term</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>
 <b>by </b><i><a class="ref" href="abcmiz_1.html#D25" target="_self" title="ABCMIZ_1:def.25">MSVARS</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c11">x'</font> = <font color="Maroon" title="c10">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K14" title="ABCMIZ_1:func.14">a_Term</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> ;<br/><b>reconsider </b><font color="Maroon" title="c12">xx</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x'</font>,<span class="p2">(<span class="default"><a href="abcmiz_1.html#K14" title="ABCMIZ_1:func.14">a_Term</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <span class="p1">(<span class="default"><a href="msafree.html#K5" title="MSAFREE:func.5">DTConMSA</a> <span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <b>by </b><i><a class="txt" href="abcmiz_1.html#E2:236_1_2"><i><font color="Green" title="E123">00</font></i></a>, <a class="ref" href="msafree3.html#T3" title="MSAFREE3:th.3">MSAFREE3:3</a></i>;<br/><a NAME="E5:236_1_2"/>
( <font color="Maroon" title="c10">x</font> <a href="abcmiz_1.html#K35" title="ABCMIZ_1:func.35">-term</a> <font color="Maroon">F<sub>1</sub></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">xx</font> &amp; <font color="Maroon" title="c12">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="abcmiz_1.html#K51" title="ABCMIZ_1:func.51">Terminals</a> <span class="p1">(<span class="default"><a href="msafree.html#K5" title="MSAFREE:func.5">DTConMSA</a> <span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="abcmiz_1.html#E2:236_1_2"><i><font color="Green" title="E123">00</font></i></a>, <a class="ref" href="msafree.html#T7" title="MSAFREE:th.7">MSAFREE:7</a></i>;<br/><b>hence </b><font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="abcmiz_1.html#K35" title="ABCMIZ_1:func.35">-term</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> = 
<font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c12">xx</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>)
<b>by </b><i><a class="txt" href="abcmiz_1.html#E2:236_1"><i><font color="Green" title="E120">A6</font></i></a></i>
<br/>.= 
<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c10">x</font>)
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">a</font> being   <a href="abcmiz_1.html#M1" title="ABCMIZ_1:mode.1">expression</a> of <font color="Maroon">F<sub>1</sub></font>(), <a href="abcmiz_1.html#K13" title="ABCMIZ_1:func.13">an_Adj</a> <font color="Maroon">F<sub>1</sub></font>() holds  <font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K32" title="ABCMIZ_1:func.32">non_op</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="abcmiz_1.html#K30" title="ABCMIZ_1:func.30">term</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font></span>)</span>) ) &amp; ( for <font color="Olive" title="b1">a</font> being   <a href="abcmiz_1.html#M1" title="ABCMIZ_1:mode.1">expression</a> of <font color="Maroon">F<sub>1</sub></font>(), <a href="abcmiz_1.html#K13" title="ABCMIZ_1:func.13">an_Adj</a> <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive" title="b2">t</font> being   <a href="abcmiz_1.html#M1" title="ABCMIZ_1:mode.1">expression</a> of <font color="Maroon">F<sub>1</sub></font>(), <a href="abcmiz_1.html#K12" title="ABCMIZ_1:func.12">a_Type</a> <font color="Maroon">F<sub>1</sub></font>() holds  <font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K33" title="ABCMIZ_1:func.33">ast</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="abcmiz_1.html#K31" title="ABCMIZ_1:func.31">term</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">t</font></span>)</span>) ) )</span>
<div class="add"><b>let </b><font color="Maroon" title="c10">c</font> be  <a href="abcmiz_1.html#V2" title="ABCMIZ_1:attr.2">constructor</a> <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon">F<sub>1</sub></font>(); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being   <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="abcmiz_1.html#K34" title="ABCMIZ_1:func.34">QuasiTerms</a> <font color="Maroon">F<sub>1</sub></font>()  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c10">c</font></span>)</span> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Olive" title="b1">p</font> holds <br/><font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">c</font> <a href="abcmiz_1.html#K36" title="ABCMIZ_1:func.36">-trm</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c10">c</font>,<font color="Olive" title="b2">q</font>)</span><br/><a NAME="E1:236_1_3"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E2:236_1_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">c</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()</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="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon">F<sub>1</sub></font>(),<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c11">cc</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">c</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <span class="p1">(<span class="default"><a href="msafree.html#K5" title="MSAFREE:func.5">DTConMSA</a> <span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><b>let </b><font color="Maroon" title="c12">p</font>, <font color="Maroon" title="c13">q</font> be    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="abcmiz_1.html#K34" title="ABCMIZ_1:func.34">QuasiTerms</a> <font color="Maroon">F<sub>1</sub></font>(); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c10">c</font></span>)</span> &amp; <font color="Maroon" title="c13">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c12">p</font> implies <font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">c</font> <a href="abcmiz_1.html#K36" title="ABCMIZ_1:func.36">-trm</a> <font color="Maroon" title="c12">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c13">q</font>) )</span><br/><b>assume </b><a NAME="E4:236_1_3"/><i><font color="Green" title="E123">Z0</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c10">c</font></span>)</span> &amp; <font color="Maroon" title="c13">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c12">p</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">c</font> <a href="abcmiz_1.html#K36" title="ABCMIZ_1:func.36">-trm</a> <font color="Maroon" title="c12">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c13">q</font>)</span><br/><b>set </b><font color="Maroon" title="c14">a</font> = <font color="Maroon" title="c10">c</font> <a href="abcmiz_1.html#K36" title="ABCMIZ_1:func.36">-trm</a> <font color="Maroon" title="c12">p</font>;<br/><b>set </b><font color="Maroon" title="c15">nt</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">c</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/><a NAME="E5:236_1_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">c</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></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="abcmiz_1.html#K52" title="ABCMIZ_1:func.52">NonTerminals</a> <span class="p1">(<span class="default"><a href="msafree.html#K5" title="MSAFREE:func.5">DTConMSA</a> <span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="abcmiz_1.html#T123" target="_self" title="ABCMIZ_1:th.123">LemNTerm</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c16">nt</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">c</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <span class="p1">(<span class="default"><a href="msafree.html#K5" title="MSAFREE:func.5">DTConMSA</a> <span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> ;<br/><b>reconsider </b><font color="Maroon" title="c17">ts</font> = <font color="Maroon" title="c12">p</font> as    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="msafree.html#K5" title="MSAFREE:func.5">DTConMSA</a> <span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <b>by </b><i><a class="txt" href="abcmiz_1.html#E4:236_1"><i><font color="Green" title="E122">A5</font></i></a></i>;<br/><a NAME="E8:236_1_3"/><i><font color="Green" title="E124">Z3</font></i>: 
<font color="Maroon" title="c10">c</font> <a href="abcmiz_1.html#K36" title="ABCMIZ_1:func.36">-trm</a> <font color="Maroon" title="c12">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c17">ts</font>
 <b>by </b><i><a class="txt" href="abcmiz_1.html#E4:236_1_3"><i><font color="Green" title="E123">Z0</font></i></a>, <a class="ref" href="abcmiz_1.html#D35" target="_self" title="ABCMIZ_1:def.35">TERM</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c18">aa</font> = <font color="Maroon" title="c10">c</font> <a href="abcmiz_1.html#K36" title="ABCMIZ_1:func.36">-trm</a> <font color="Maroon" title="c12">p</font> as   <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon">F<sub>1</sub></font>(),<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="msafree3.html#T9" title="MSAFREE3:th.9">MSAFREE3:9</a></i>;<br/><a NAME="E10:236_1_3"/><i><font color="Green" title="E125">Z4</font></i>: 
( <font color="Maroon" title="c16">nt</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">c</font> &amp; <font color="Maroon" title="c10">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="abcmiz_1.html#K9" title="ABCMIZ_1:func.9">*</a>  &amp; <font color="Maroon" title="c10">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="abcmiz_1.html#K10" title="ABCMIZ_1:func.10">non_op</a>  )
 <b>by </b><i><a class="ref" href="abcmiz_1.html#D11" target="_self" title="ABCMIZ_1:def.11">CNSTR2</a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/><a NAME="E11:236_1_3"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c12">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>1</sub></font>() <a href="msaterm.html#K1" title="MSATERM:func.1">-Terms</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="236_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c12">p</font> or <font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() <a href="msaterm.html#K1" title="MSATERM:func.1">-Terms</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:236_1_3_1"/>
<font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c12">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() <a href="msaterm.html#K1" title="MSATERM:func.1">-Terms</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span><br/>

<a NAME="E2:236_1_3_1"/><b>then </b>
<font color="Maroon" title="c19">z</font> is    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="msaterm.html#K1" title="MSATERM:func.1">-Terms</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="msafree3.html#T9" title="MSAFREE3:th.9">MSAFREE3:9</a></i>;<br/>
<b>hence </b><a NAME="E3:236_1_3_1"/>
<font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() <a href="msaterm.html#K1" title="MSATERM:func.1">-Terms</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c19">r</font> = <font color="Maroon" title="c12">p</font> as    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="msaterm.html#K1" title="MSATERM:func.1">-Terms</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/><a NAME="E13:236_1_3"/>
<font color="Maroon" title="c10">c</font> <a href="abcmiz_1.html#K36" title="ABCMIZ_1:func.36">-trm</a> <font color="Maroon" title="c12">p</font> is   <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon">F<sub>1</sub></font>(),<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="msafree3.html#T9" title="MSAFREE3:th.9">MSAFREE3:9</a></i>;<br/><a NAME="E14:236_1_3"/><b>then </b>
<font color="Maroon" title="c19">r</font> is    <a href="msaterm.html#M1" title="MSATERM:mode.1">ArgumentSeq</a> of  <a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> <font color="Maroon" title="c10">c</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="abcmiz_1.html#E8:236_1_3"><i><font color="Green" title="E124">Z3</font></i></a>, <a class="ref" href="msaterm.html#T1" title="MSATERM:th.1">MSATERM:1</a></i>;<br/><a NAME="E15:236_1_3"/><b>then </b>
 <a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> <font color="Maroon" title="c10">c</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c19">r</font>
 <b>by </b><i><a class="ref" href="msaterm.html#T21" title="MSATERM:th.21">MSATERM:21</a></i>;<br/><a NAME="E16:236_1_3"/><b>then </b>
<font color="Maroon" title="c16">nt</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c17">ts</font>
 <b>by </b><i><a class="ref" href="abcmiz_1.html#T127" target="_self" title="ABCMIZ_1:th.127">LemArr</a></i>;<br/><b>then </b><font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">c</font> <a href="abcmiz_1.html#K36" title="ABCMIZ_1:func.36">-trm</a> <font color="Maroon" title="c12">p</font></span>)</span> = 
<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c16">nt</font>, <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c17">ts</font>,<font color="Maroon" title="c9">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c17">ts</font>)
<b>by </b><i><a class="txt" href="abcmiz_1.html#E3:236_1"><i><font color="Green" title="E121">A7</font></i></a>, <a class="txt" href="abcmiz_1.html#E8:236_1_3"><i><font color="Green" title="E124">Z3</font></i></a></i>
<br/>.= 
 <a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> <font color="Maroon" title="c10">c</font>,<a href="abcmiz_1.html#K10" title="ABCMIZ_1:func.10">non_op</a> ,<font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c17">ts</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>),<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c10">c</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c17">ts</font></span>)</span>)
<b>by </b><i><a class="txt" href="abcmiz_1.html#E10:236_1_3"><i><font color="Green" title="E125">Z4</font></i></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>
<br/>.= 
<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c10">c</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c17">ts</font></span>)</span>)
<b>by </b><i><a class="txt" href="abcmiz_1.html#E10:236_1_3"><i><font color="Green" title="E125">Z4</font></i></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>
;<br/><b>hence </b><a NAME="E18:236_1_3"/>
<font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">c</font> <a href="abcmiz_1.html#K36" title="ABCMIZ_1:func.36">-trm</a> <font color="Maroon" title="c12">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c13">q</font>)
 <b>by </b><i><a class="txt" href="abcmiz_1.html#E4:236_1_3"><i><font color="Green" title="E123">Z0</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font> being   <a href="abcmiz_1.html#M1" title="ABCMIZ_1:mode.1">expression</a> of <font color="Maroon">F<sub>1</sub></font>(), <a href="abcmiz_1.html#K13" title="ABCMIZ_1:func.13">an_Adj</a> <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive" title="b2">t</font> being   <a href="abcmiz_1.html#M1" title="ABCMIZ_1:mode.1">expression</a> of <font color="Maroon">F<sub>1</sub></font>(), <a href="abcmiz_1.html#K12" title="ABCMIZ_1:func.12">a_Type</a> <font color="Maroon">F<sub>1</sub></font>() holds  <font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K33" title="ABCMIZ_1:func.33">ast</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="abcmiz_1.html#K31" title="ABCMIZ_1:func.31">term</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">t</font></span>)</span>)</span>
<div class="add"><b>let </b><font color="Maroon" title="c10">v</font> be    <a href="abcmiz_1.html#M1" title="ABCMIZ_1:mode.1">expression</a> of <font color="Maroon">F<sub>1</sub></font>(), <a href="abcmiz_1.html#K13" title="ABCMIZ_1:func.13">an_Adj</a> <font color="Maroon">F<sub>1</sub></font>(); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K32" title="ABCMIZ_1:func.32">non_op</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="abcmiz_1.html#K30" title="ABCMIZ_1:func.30">term</a> <font color="Maroon" title="c10">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">v</font></span>)</span>)</span><br/><a NAME="E1:236_1_4"/><i><font color="Green" title="E123">Y1</font></i>: 
<font color="Maroon" title="c10">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K13" title="ABCMIZ_1:func.13">an_Adj</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>
 <b>by </b><i><a class="ref" href="abcmiz_1.html#D28" target="_self" title="ABCMIZ_1:def.28">ELEMENT</a></i>;<br/><a NAME="E2:236_1_4"/><b>then </b><i><font color="Green" title="E124">Y2</font></i>: 
<font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K13" title="ABCMIZ_1:func.13">an_Adj</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>
 <b>by </b><i><a class="ref" href="abcmiz_1.html#T129" target="_self" title="ABCMIZ_1:th.129">ThTr</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c11">fv</font> = <font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">v</font> as    <a href="abcmiz_1.html#M1" title="ABCMIZ_1:mode.1">expression</a> of <font color="Maroon">F<sub>1</sub></font>(), <a href="abcmiz_1.html#K13" title="ABCMIZ_1:func.13">an_Adj</a> <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="ref" href="abcmiz_1.html#D28" target="_self" title="ABCMIZ_1:def.28">ELEMENT</a></i>;<br/><b>set </b><font color="Maroon" title="c12">QA</font> = the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K13" title="ABCMIZ_1:func.13">an_Adj</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>;<br/><a NAME="E4:236_1_4"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c10">v</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T55" title="FINSEQ_1:th.55">FINSEQ_1:55</a></i>;<br/><a NAME="E5:236_1_4"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c10">v</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K13" title="ABCMIZ_1:func.13">an_Adj</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>
 <b>by </b><i><a class="txt" href="abcmiz_1.html#E1:236_1_4"><i><font color="Green" title="E123">Y1</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c13">p</font> = <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c10">v</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K13" title="ABCMIZ_1:func.13">an_Adj</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/><b>set </b><font color="Maroon" title="c14">c</font> =  <a href="abcmiz_1.html#K32" title="ABCMIZ_1:func.32">non_op</a> <font color="Maroon">F<sub>1</sub></font>();<br/><b>set </b><font color="Maroon" title="c15">a</font> = <span class="p1">(<span class="default"><a href="abcmiz_1.html#K32" title="ABCMIZ_1:func.32">non_op</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="abcmiz_1.html#K30" title="ABCMIZ_1:func.30">term</a> <font color="Maroon" title="c10">v</font>;<br/><b>set </b><font color="Maroon" title="c16">nt</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K32" title="ABCMIZ_1:func.32">non_op</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/><a NAME="E7:236_1_4"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K32" title="ABCMIZ_1:func.32">non_op</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></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="abcmiz_1.html#K52" title="ABCMIZ_1:func.52">NonTerminals</a> <span class="p1">(<span class="default"><a href="msafree.html#K5" title="MSAFREE:func.5">DTConMSA</a> <span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="abcmiz_1.html#T123" target="_self" title="ABCMIZ_1:th.123">LemNTerm</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c17">nt</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K32" title="ABCMIZ_1:func.32">non_op</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <span class="p1">(<span class="default"><a href="msafree.html#K5" title="MSAFREE:func.5">DTConMSA</a> <span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> ;<br/><b>reconsider </b><font color="Maroon" title="c18">ts</font> = <font color="Maroon" title="c13">p</font> as    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="msafree.html#K5" title="MSAFREE:func.5">DTConMSA</a> <span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <b>by </b><i><a class="txt" href="abcmiz_1.html#E4:236_1"><i><font color="Green" title="E122">A5</font></i></a></i>;<br/><a NAME="E10:236_1_4"/><i><font color="Green" title="E125">Z3</font></i>: 
<span class="p1">(<span class="default"><a href="abcmiz_1.html#K32" title="ABCMIZ_1:func.32">non_op</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="abcmiz_1.html#K30" title="ABCMIZ_1:func.30">term</a> <font color="Maroon" title="c10">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c18">ts</font>
 <b>by </b><i><a class="ref" href="abcmiz_1.html#T43" target="_self" title="ABCMIZ_1:th.43">ThNon</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c19">aa</font> = <span class="p1">(<span class="default"><a href="abcmiz_1.html#K32" title="ABCMIZ_1:func.32">non_op</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="abcmiz_1.html#K30" title="ABCMIZ_1:func.30">term</a> <font color="Maroon" title="c10">v</font> as   <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon">F<sub>1</sub></font>(),<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="msafree3.html#T9" title="MSAFREE3:th.9">MSAFREE3:9</a></i>;<br/><a NAME="E12:236_1_4"/><i><font color="Green" title="E126">Z4</font></i>: 
( <font color="Maroon" title="c17">nt</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="abcmiz_1.html#K32" title="ABCMIZ_1:func.32">non_op</a> <font color="Maroon">F<sub>1</sub></font>() &amp;  <a href="abcmiz_1.html#K32" title="ABCMIZ_1:func.32">non_op</a> <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="abcmiz_1.html#K9" title="ABCMIZ_1:func.9">*</a>  )
 <b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/><a NAME="E13:236_1_4"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E14:236_1_4"/><b>then </b><i><font color="Green" title="E127">08</font></i>: 
<font color="Maroon" title="c9">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c13">p</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c11">fv</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="bagorder.html#T3" title="BAGORDER:th.3">BAGORDER:3</a></i>;<br/><a NAME="E15:236_1_4"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c11">fv</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c11">fv</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T55" title="FINSEQ_1:th.55">FINSEQ_1:55</a></i>;<br/><a NAME="E16:236_1_4"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c11">fv</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K13" title="ABCMIZ_1:func.13">an_Adj</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>
 <b>by </b><i><a class="txt" href="abcmiz_1.html#E2:236_1_4"><i><font color="Green" title="E124">Y2</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c20">q</font> = <font color="Maroon" title="c9">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c13">p</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K13" title="ABCMIZ_1:func.13">an_Adj</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <b>by </b><i><a class="txt" href="abcmiz_1.html#E14:236_1_4"><i><font color="Green" title="E127">08</font></i></a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/><a NAME="E18:236_1_4"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c13">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>1</sub></font>() <a href="msaterm.html#K1" title="MSATERM:func.1">-Terms</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="236_1_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c21">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c21">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c13">p</font> or <font color="Maroon" title="c21">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() <a href="msaterm.html#K1" title="MSATERM:func.1">-Terms</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:236_1_4_1"/>
<font color="Maroon" title="c21">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c13">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c21">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() <a href="msaterm.html#K1" title="MSATERM:func.1">-Terms</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span><br/>

<a NAME="E2:236_1_4_1"/><b>then </b>
<font color="Maroon" title="c21">z</font> is    <a href="abcmiz_1.html#M1" title="ABCMIZ_1:mode.1">expression</a> of <font color="Maroon">F<sub>1</sub></font>(), <a href="abcmiz_1.html#K13" title="ABCMIZ_1:func.13">an_Adj</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="ref" href="abcmiz_1.html#T41" target="_self" title="ABCMIZ_1:th.41">Th42</a></i>;<br/>
<a NAME="E3:236_1_4_1"/><b>then </b>
<font color="Maroon" title="c21">z</font> is    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="msaterm.html#K1" title="MSATERM:func.1">-Terms</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="msafree3.html#T9" title="MSAFREE3:th.9">MSAFREE3:9</a></i>;<br/>
<b>hence </b><a NAME="E4:236_1_4_1"/>
<font color="Maroon" title="c21">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() <a href="msaterm.html#K1" title="MSATERM:func.1">-Terms</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c21">r</font> = <font color="Maroon" title="c13">p</font> as    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="msaterm.html#K1" title="MSATERM:func.1">-Terms</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/><a NAME="E20:236_1_4"/>
<span class="p1">(<span class="default"><a href="abcmiz_1.html#K32" title="ABCMIZ_1:func.32">non_op</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="abcmiz_1.html#K30" title="ABCMIZ_1:func.30">term</a> <font color="Maroon" title="c10">v</font> is   <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon">F<sub>1</sub></font>(),<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="msafree3.html#T9" title="MSAFREE3:th.9">MSAFREE3:9</a></i>;<br/><a NAME="E21:236_1_4"/><b>then </b>
<font color="Maroon" title="c21">r</font> is    <a href="msaterm.html#M1" title="MSATERM:mode.1">ArgumentSeq</a> of  <a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K32" title="ABCMIZ_1:func.32">non_op</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="abcmiz_1.html#E10:236_1_4"><i><font color="Green" title="E125">Z3</font></i></a>, <a class="ref" href="msaterm.html#T1" title="MSATERM:th.1">MSATERM:1</a></i>;<br/><a NAME="E22:236_1_4"/><b>then </b>
 <a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K32" title="ABCMIZ_1:func.32">non_op</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c21">r</font>
 <b>by </b><i><a class="ref" href="msaterm.html#T21" title="MSATERM:th.21">MSATERM:21</a></i>;<br/><a NAME="E23:236_1_4"/><b>then </b>
<font color="Maroon" title="c17">nt</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c18">ts</font>
 <b>by </b><i><a class="ref" href="abcmiz_1.html#T127" target="_self" title="ABCMIZ_1:th.127">LemArr</a></i>;<br/><b>then </b><font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K32" title="ABCMIZ_1:func.32">non_op</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="abcmiz_1.html#K30" title="ABCMIZ_1:func.30">term</a> <font color="Maroon" title="c10">v</font></span>)</span> = 
<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c17">nt</font>, <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c18">ts</font>,<font color="Maroon" title="c9">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c18">ts</font>)
<b>by </b><i><a class="txt" href="abcmiz_1.html#E3:236_1"><i><font color="Green" title="E121">A7</font></i></a>, <a class="txt" href="abcmiz_1.html#E10:236_1_4"><i><font color="Green" title="E125">Z3</font></i></a></i>
<br/>.= 
 <a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K32" title="ABCMIZ_1:func.32">non_op</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,<a href="abcmiz_1.html#K10" title="ABCMIZ_1:func.10">non_op</a> ,<font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c18">ts</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>),<font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><a href="abcmiz_1.html#K32" title="ABCMIZ_1:func.32">non_op</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c18">ts</font></span>)</span>)
<b>by </b><i><a class="txt" href="abcmiz_1.html#E12:236_1_4"><i><font color="Green" title="E126">Z4</font></i></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>
<br/>.= 
<font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c18">ts</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>)
<b>by </b><i><a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>
;<br/><b>hence </b><a NAME="E25:236_1_4"/>
<font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K32" title="ABCMIZ_1:func.32">non_op</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="abcmiz_1.html#K30" title="ABCMIZ_1:func.30">term</a> <font color="Maroon" title="c10">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">v</font></span>)</span>)
 <b>by </b><i><a class="txt" href="abcmiz_1.html#E14:236_1_4"><i><font color="Green" title="E127">08</font></i></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c10">v</font> be    <a href="abcmiz_1.html#M1" title="ABCMIZ_1:mode.1">expression</a> of <font color="Maroon">F<sub>1</sub></font>(), <a href="abcmiz_1.html#K13" title="ABCMIZ_1:func.13">an_Adj</a> <font color="Maroon">F<sub>1</sub></font>(); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">t</font> being   <a href="abcmiz_1.html#M1" title="ABCMIZ_1:mode.1">expression</a> of <font color="Maroon">F<sub>1</sub></font>(), <a href="abcmiz_1.html#K12" title="ABCMIZ_1:func.12">a_Type</a> <font color="Maroon">F<sub>1</sub></font>() holds  <font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K33" title="ABCMIZ_1:func.33">ast</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="abcmiz_1.html#K31" title="ABCMIZ_1:func.31">term</a> <font color="Maroon" title="c10">v</font>,<font color="Olive" title="b1">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">v</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">t</font></span>)</span>)</span><br/><b>let </b><font color="Maroon" title="c11">t</font> be    <a href="abcmiz_1.html#M1" title="ABCMIZ_1:mode.1">expression</a> of <font color="Maroon">F<sub>1</sub></font>(), <a href="abcmiz_1.html#K12" title="ABCMIZ_1:func.12">a_Type</a> <font color="Maroon">F<sub>1</sub></font>(); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K33" title="ABCMIZ_1:func.33">ast</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="abcmiz_1.html#K31" title="ABCMIZ_1:func.31">term</a> <font color="Maroon" title="c10">v</font>,<font color="Maroon" title="c11">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">v</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">t</font></span>)</span>)</span><br/>



<a NAME="E11:236_1"/>
( <font color="Maroon" title="c10">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K13" title="ABCMIZ_1:func.13">an_Adj</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> &amp; <font color="Maroon" title="c11">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K12" title="ABCMIZ_1:func.12">a_Type</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> )
 
<b>by </b><i><a class="ref" href="abcmiz_1.html#D28" target="_self" title="ABCMIZ_1:def.28">ELEMENT</a></i>;<br/>
<a NAME="E12:236_1"/><b>then </b><i><font color="Green" title="E123">Y2</font></i>: 
( <font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K13" title="ABCMIZ_1:func.13">an_Adj</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> &amp; <font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K12" title="ABCMIZ_1:func.12">a_Type</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> )
 
<b>by </b><i><a class="ref" href="abcmiz_1.html#T129" target="_self" title="ABCMIZ_1:th.129">ThTr</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">fv</font> = <font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">v</font> as    <a href="abcmiz_1.html#M1" title="ABCMIZ_1:mode.1">expression</a> of <font color="Maroon">F<sub>1</sub></font>(), <a href="abcmiz_1.html#K13" title="ABCMIZ_1:func.13">an_Adj</a> <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="ref" href="abcmiz_1.html#D28" target="_self" title="ABCMIZ_1:def.28">ELEMENT</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">ft</font> = <font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">t</font> as    <a href="abcmiz_1.html#M1" title="ABCMIZ_1:mode.1">expression</a> of <font color="Maroon">F<sub>1</sub></font>(), <a href="abcmiz_1.html#K12" title="ABCMIZ_1:func.12">a_Type</a> <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="txt" href="abcmiz_1.html#E12:236_1"><i><font color="Green" title="E123">Y2</font></i></a>, <a class="ref" href="abcmiz_1.html#D28" target="_self" title="ABCMIZ_1:def.28">ELEMENT</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">p</font> = <span class="p1"><a href="lang1.html#K4" title="LANG1:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c10">v</font>,<font color="Maroon" title="c11">t</font></span><a href="lang1.html#K4" title="LANG1:func.4">*&gt;</a></span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> ;<br/>
<b>set </b><font color="Maroon" title="c15">c</font> =  <a href="abcmiz_1.html#K33" title="ABCMIZ_1:func.33">ast</a> <font color="Maroon">F<sub>1</sub></font>();<br/>
<b>set </b><font color="Maroon" title="c16">a</font> = <span class="p1">(<span class="default"><a href="abcmiz_1.html#K33" title="ABCMIZ_1:func.33">ast</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="abcmiz_1.html#K31" title="ABCMIZ_1:func.31">term</a> <font color="Maroon" title="c10">v</font>,<font color="Maroon" title="c11">t</font>;<br/>
<b>set </b><font color="Maroon" title="c17">nt</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K33" title="ABCMIZ_1:func.33">ast</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>
<a NAME="E16:236_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K33" title="ABCMIZ_1:func.33">ast</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></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="abcmiz_1.html#K52" title="ABCMIZ_1:func.52">NonTerminals</a> <span class="p1">(<span class="default"><a href="msafree.html#K5" title="MSAFREE:func.5">DTConMSA</a> <span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="abcmiz_1.html#T123" target="_self" title="ABCMIZ_1:th.123">LemNTerm</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c18">nt</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K33" title="ABCMIZ_1:func.33">ast</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <span class="p1">(<span class="default"><a href="msafree.html#K5" title="MSAFREE:func.5">DTConMSA</a> <span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c19">ts</font> = <font color="Maroon" title="c14">p</font> as    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="msafree.html#K5" title="MSAFREE:func.5">DTConMSA</a> <span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <b>by </b><i><a class="txt" href="abcmiz_1.html#E4:236_1"><i><font color="Green" title="E122">A5</font></i></a></i>;<br/>
<a NAME="E19:236_1"/><i><font color="Green" title="E124">Z3</font></i>: 
<span class="p1">(<span class="default"><a href="abcmiz_1.html#K33" title="ABCMIZ_1:func.33">ast</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="abcmiz_1.html#K31" title="ABCMIZ_1:func.31">term</a> <font color="Maroon" title="c10">v</font>,<font color="Maroon" title="c11">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c19">ts</font>
 
<b>by </b><i><a class="ref" href="abcmiz_1.html#T46" target="_self" title="ABCMIZ_1:th.46">ThAst</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c20">aa</font> = <span class="p1">(<span class="default"><a href="abcmiz_1.html#K33" title="ABCMIZ_1:func.33">ast</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="abcmiz_1.html#K31" title="ABCMIZ_1:func.31">term</a> <font color="Maroon" title="c10">v</font>,<font color="Maroon" title="c11">t</font> as   <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon">F<sub>1</sub></font>(),<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="msafree3.html#T9" title="MSAFREE3:th.9">MSAFREE3:9</a></i>;<br/>
<a NAME="E21:236_1"/><i><font color="Green" title="E125">Z4</font></i>: 
( <font color="Maroon" title="c18">nt</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="abcmiz_1.html#K33" title="ABCMIZ_1:func.33">ast</a> <font color="Maroon">F<sub>1</sub></font>() &amp;  <a href="abcmiz_1.html#K33" title="ABCMIZ_1:func.33">ast</a> <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="abcmiz_1.html#K10" title="ABCMIZ_1:func.10">non_op</a>  )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>
<a NAME="E22:236_1"/><i><font color="Green" title="E126">08</font></i>: 
<font color="Maroon" title="c9">f</font> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c14">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="lang1.html#K4" title="LANG1:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c12">fv</font>,<font color="Maroon" title="c13">ft</font></span><a href="lang1.html#K4" title="LANG1:func.4">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T40" title="FINSEQ_2:th.40">FINSEQ_2:40</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c21">q</font> = <font color="Maroon" title="c9">f</font> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c14">p</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> ;<br/>
<a NAME="E24:236_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>1</sub></font>() <a href="msaterm.html#K1" title="MSATERM:func.1">-Terms</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="236_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c22">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c22">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">p</font> or <font color="Maroon" title="c22">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() <a href="msaterm.html#K1" title="MSATERM:func.1">-Terms</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:236_1_5"/>
<font color="Maroon" title="c22">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c22">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() <a href="msaterm.html#K1" title="MSATERM:func.1">-Terms</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span><br/>

<a NAME="E2:236_1_5"/><b>then </b>
<font color="Maroon" title="c22">z</font> is    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="msaterm.html#K1" title="MSATERM:func.1">-Terms</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="msafree3.html#T9" title="MSAFREE3:th.9">MSAFREE3:9</a></i>;<br/>
<b>hence </b><a NAME="E3:236_1_5"/>
<font color="Maroon" title="c22">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() <a href="msaterm.html#K1" title="MSATERM:func.1">-Terms</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c22">r</font> = <font color="Maroon" title="c14">p</font> as    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="msaterm.html#K1" title="MSATERM:func.1">-Terms</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E26:236_1"/>
<span class="p1">(<span class="default"><a href="abcmiz_1.html#K33" title="ABCMIZ_1:func.33">ast</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="abcmiz_1.html#K31" title="ABCMIZ_1:func.31">term</a> <font color="Maroon" title="c10">v</font>,<font color="Maroon" title="c11">t</font> is   <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon">F<sub>1</sub></font>(),<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="msafree3.html#T9" title="MSAFREE3:th.9">MSAFREE3:9</a></i>;<br/>
<a NAME="E27:236_1"/><b>then </b>
<font color="Maroon" title="c22">r</font> is    <a href="msaterm.html#M1" title="MSATERM:mode.1">ArgumentSeq</a> of  <a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K33" title="ABCMIZ_1:func.33">ast</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="abcmiz_1.html#E19:236_1"><i><font color="Green" title="E124">Z3</font></i></a>, <a class="ref" href="msaterm.html#T1" title="MSATERM:th.1">MSATERM:1</a></i>;<br/>
<a NAME="E28:236_1"/><b>then </b>
 <a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K33" title="ABCMIZ_1:func.33">ast</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c22">r</font>
 
<b>by </b><i><a class="ref" href="msaterm.html#T21" title="MSATERM:th.21">MSATERM:21</a></i>;<br/>
<a NAME="E29:236_1"/><b>then </b>
<font color="Maroon" title="c18">nt</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c19">ts</font>
 
<b>by </b><i><a class="ref" href="abcmiz_1.html#T127" target="_self" title="ABCMIZ_1:th.127">LemArr</a></i>;<br/>
<b>then </b><font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K33" title="ABCMIZ_1:func.33">ast</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="abcmiz_1.html#K31" title="ABCMIZ_1:func.31">term</a> <font color="Maroon" title="c10">v</font>,<font color="Maroon" title="c11">t</font></span>)</span> = 
<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c18">nt</font>, <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c19">ts</font>,<font color="Maroon" title="c9">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c19">ts</font>)
<b>by </b><i><a class="txt" href="abcmiz_1.html#E3:236_1"><i><font color="Green" title="E121">A7</font></i></a>, <a class="txt" href="abcmiz_1.html#E19:236_1"><i><font color="Green" title="E124">Z3</font></i></a></i>
<br/>.= 
<font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c19">ts</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c19">ts</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</span>)</span>)
<b>by </b><i><a class="txt" href="abcmiz_1.html#E21:236_1"><i><font color="Green" title="E125">Z4</font></i></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>
<br/>.= 
<font color="Maroon">F<sub>5</sub></font>(<font color="Maroon" title="c12">fv</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c19">ts</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</span>)</span>)
<b>by </b><i><a class="txt" href="abcmiz_1.html#E22:236_1"><i><font color="Green" title="E126">08</font></i></a>, <a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>
;<br/>
<b>hence </b><a NAME="E31:236_1"/>
<font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K33" title="ABCMIZ_1:func.33">ast</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="abcmiz_1.html#K31" title="ABCMIZ_1:func.31">term</a> <font color="Maroon" title="c10">v</font>,<font color="Maroon" title="c11">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">v</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">t</font></span>)</span>)
 <b>by </b><i><a class="txt" href="abcmiz_1.html#E22:236_1"><i><font color="Green" title="E126">08</font></i></a>, <a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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