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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( $1 is   <a href="abcmiz_1.html#NM6" title="ABCMIZ_1:NM.6">expression</a> of <font color="Maroon">F<sub>1</sub></font>() implies <font color="Maroon">P<sub>1</sub></font>[$1] );<br/>
<b>set </b><font color="Maroon" title="c1">X</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">V</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">S</font> = <font color="Maroon">F<sub>1</sub></font>();<br/>
<b>set </b><font color="Maroon" title="c4">C</font> = <font color="Maroon">F<sub>1</sub></font>();<br/>
<a NAME="E1:210_1"/><i><font color="Green" title="E104">B0</font></i>: 
<font color="Maroon">F<sub>2</sub></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="E2:210_1"/><i><font color="Green" title="E105">B1</font></i>: 
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>()<br/> for <font color="Olive" title="b2">v</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font> holds  <font color="Maroon">S<sub>1</sub></font>[ <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b2">v</font>,<font color="Olive" title="b1">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="210_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">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"> for <font color="Olive" title="b1">v</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s</font> holds  <font color="Maroon">S<sub>1</sub></font>[ <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Maroon" title="c5">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>]</span><br/><b>let </b><font color="Maroon" title="c6">v</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s</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>[ <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c5">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>]</span><br/>



<b>set </b><font color="Maroon" title="c7">t</font> =  <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c5">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>;<br/>
<b>assume </b><a NAME="E1:210_1_1"/><i><font color="Green" title="E106">C1</font></i>: 
 <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c5">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> is   <a href="abcmiz_1.html#NM6" title="ABCMIZ_1:NM.6">expression</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">P<sub>1</sub></font>[ <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c5">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>]</span><br/>

<a NAME="E2:210_1_1"/>
( <span class="p1">(<span class="default"><a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c5">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c5">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Maroon" title="c5">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() )
 
<b>by </b><i><a class="ref" href="trees_4.html#T3" title="TREES_4:th.3">TREES_4:3</a></i>;<br/>
<a NAME="E3:210_1_1"/><b>then </b><i><font color="Green" title="E107">C2</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c5">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s</font> &amp; <font color="Maroon" title="c5">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="210_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:210_1_1_1"/>
(  ex <font color="Olive" title="b1">x</font> being  <a href="abcmiz_1.html#NM3" title="ABCMIZ_1:NM.3">variable</a> st  <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c5">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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>() or  ex <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>() ex <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 <br/>(  <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;  <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c5">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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> ) or  ex <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  <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c5">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#K30" title="ABCMIZ_1:func.30">term</a> <font color="Olive" title="b1">a</font> or  ex <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>() ex <font color="Olive" title="b2">q</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  <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c5">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="abcmiz_1.html#K31" title="ABCMIZ_1:func.31">term</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">q</font> )
 </span><b>by </b><i><a class="txt" href="abcmiz_1.html#E1:210_1_1"><i><font color="Green" title="E106">C1</font></i></a>, <a class="ref" href="abcmiz_1.html#T53" target="_self" title="ABCMIZ_1:th.53">Th100</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="210_1_1_1_1"><b>suppose </b></a><a NAME="E1:210_1_1_1_1"/>
 ex <font color="Olive" title="b1">x</font> being  <a href="abcmiz_1.html#NM3" title="ABCMIZ_1:NM.3">variable</a> st  <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c5">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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>()
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[ <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c5">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>]</span><br/><div class="add"><b>hence </b><a NAME="E2:210_1_1_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[ <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c5">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>]
 <b>by </b><i><a class="txt" href="abcmiz_1.html#E1:210"><i><font color="Green" title="E100">W1</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="210_1_1_1_2"><b>suppose </b></a><a NAME="E1:210_1_1_1_2"/>
 ex <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>() ex <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 <br/>(  <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;  <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c5">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[ <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c5">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>]</span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c8">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>(), <font color="Maroon" title="c9">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>()<b> such that </b><br/><a NAME="E3:210_1_1_1_2"/><i><font color="Green" title="E108">C3</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">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="c8">c</font></span>)</span> &amp;  <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c5">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c</font> <a href="abcmiz_1.html#K36" title="ABCMIZ_1:func.36">-trm</a> <font color="Maroon" title="c9">p</font> )
 ;<br/><a NAME="E4:210_1_1_1_2"/>
 <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c5">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">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="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c9">p</font>
 <b>by </b><i><a class="txt" href="abcmiz_1.html#E3:210_1_1_1_2"><i><font color="Green" title="E108">C3</font></i></a>, <a class="ref" href="abcmiz_1.html#D35" target="_self" title="ABCMIZ_1:def.35">TERM</a></i>;<br/><a NAME="E5:210_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c5">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">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>
 <b>by </b><i><a class="ref" href="trees_4.html#D4" title="TREES_4:def.4">TREES_4:def 4</a></i>;<br/><b>hence </b><a NAME="E6:210_1_1_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[ <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c5">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>]
 <b>by </b><i><a class="txt" href="abcmiz_1.html#E3:210_1_1"><i><font color="Green" title="E107">C2</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="210_1_1_1_3"><b>suppose </b></a><a NAME="E1:210_1_1_1_3"/>
 ex <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  <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c5">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#K30" title="ABCMIZ_1:func.30">term</a> <font color="Olive" title="b1">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[ <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c5">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>]</span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c8">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>()<b> such that </b><br/><a NAME="E3:210_1_1_1_3"/><i><font color="Green" title="E108">C4</font></i>: 
 <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c5">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#K30" title="ABCMIZ_1:func.30">term</a> <font color="Maroon" title="c8">a</font>
 ;<br/><a NAME="E4:210_1_1_1_3"/>
(  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_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> <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><span class="p2">(<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></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> &amp; <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><span class="p2">(<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></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <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>() &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><span class="p2">(<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></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="ref" href="abcmiz_1.html#D9" target="_self" title="ABCMIZ_1:def.9">CONSTRSIGN</a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>;<br/><a NAME="E5:210_1_1_1_3"/><b>then </b>
 <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c5">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><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="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c8">a</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="abcmiz_1.html#E3:210_1_1_1_3"><i><font color="Green" title="E108">C4</font></i></a>, <a class="ref" href="abcmiz_1.html#D30" target="_self" title="ABCMIZ_1:def.30">TERM1</a></i>;<br/><a NAME="E6:210_1_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c5">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><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>
 <b>by </b><i><a class="ref" href="trees_4.html#D4" title="TREES_4:def.4">TREES_4:def 4</a></i>;<br/><b>hence </b><a NAME="E7:210_1_1_1_3"/>
<font color="Maroon">P<sub>1</sub></font>[ <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c5">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>]
 <b>by </b><i><a class="txt" href="abcmiz_1.html#E3:210_1_1"><i><font color="Green" title="E107">C2</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="210_1_1_1_4"><b>suppose </b></a><a NAME="E1:210_1_1_1_4"/>
 ex <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>() ex <font color="Olive" title="b2">q</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  <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c5">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="abcmiz_1.html#K31" title="ABCMIZ_1:func.31">term</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[ <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c5">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>]</span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c8">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>(), <font color="Maroon" title="c9">q</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>()<b> such that </b><br/><a NAME="E3:210_1_1_1_4"/><i><font color="Green" title="E108">C5</font></i>: 
 <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c5">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="abcmiz_1.html#K31" title="ABCMIZ_1:func.31">term</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">q</font>
 ;<br/><a NAME="E4:210_1_1_1_4"/>
(  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_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> <span class="p1"><a href="lang1.html#K4" title="LANG1:func.4">&lt;*</a><span class="default"><span class="p2">(<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>,<span class="p2">(<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></span><a href="lang1.html#K4" title="LANG1:func.4">*&gt;</a></span> &amp; <span class="p1"><a href="lang1.html#K4" title="LANG1:func.4">&lt;*</a><span class="default"><span class="p2">(<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>,<span class="p2">(<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></span><a href="lang1.html#K4" title="LANG1:func.4">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <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>() &amp; <span class="p1"><a href="lang1.html#K4" title="LANG1:func.4">&lt;*</a><span class="default"><span class="p2">(<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>,<span class="p2">(<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></span><a href="lang1.html#K4" title="LANG1:func.4">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <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>() &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="lang1.html#K4" title="LANG1:func.4">&lt;*</a><span class="default"><span class="p2">(<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>,<span class="p2">(<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></span><a href="lang1.html#K4" title="LANG1:func.4">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 <b>by </b><i><a class="ref" href="abcmiz_1.html#D9" target="_self" title="ABCMIZ_1:def.9">CONSTRSIGN</a>, <a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>;<br/><a NAME="E5:210_1_1_1_4"/><b>then </b>
 <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c5">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><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="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <span class="p1"><a href="lang1.html#K4" title="LANG1:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">q</font></span><a href="lang1.html#K4" title="LANG1:func.4">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="abcmiz_1.html#E3:210_1_1_1_4"><i><font color="Green" title="E108">C5</font></i></a>, <a class="ref" href="abcmiz_1.html#D31" target="_self" title="ABCMIZ_1:def.31">TERM2</a></i>;<br/><a NAME="E6:210_1_1_1_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c5">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><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>
 <b>by </b><i><a class="ref" href="trees_4.html#D4" title="TREES_4:def.4">TREES_4:def 4</a></i>;<br/><b>hence </b><a NAME="E7:210_1_1_1_4"/>
<font color="Maroon">P<sub>1</sub></font>[ <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c5">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>]
 <b>by </b><i><a class="txt" href="abcmiz_1.html#E3:210_1_1"><i><font color="Green" title="E107">C2</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</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>end;</b></div>

</div><b>end;</b></div>
<a NAME="E3:210_1"/><i><font color="Green" title="E106">B2</font></i>: 
for <font color="Olive" title="b1">o</font> being  <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="msaterm.html#M1" title="MSATERM:mode.1">ArgumentSeq</a> of  <a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> <font color="Olive" title="b1">o</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>  st ( for <font color="Olive" title="b3">t</font> being  <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>  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>[<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">o</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="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Olive" title="b2">p</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="210_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">o</font> be   <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="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="c5">o</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>  st ( for <font color="Olive" title="b2">t</font> being  <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>  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>[<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">o</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="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Olive" title="b1">p</font>]</span><br/><b>let </b><font color="Maroon" title="c6">p</font> be    <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="c5">o</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 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="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>  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="c6">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>[<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">o</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="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c6">p</font>] )</span><br/>



<b>assume </b><a NAME="E1:210_1_2"/><i><font color="Green" title="E107">Z0</font></i>: 
for <font color="Olive" title="b1">t</font> being  <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>  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="c6">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>[<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">o</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="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c6">p</font>]</span><br/>

<b>set </b><font color="Maroon" title="c7">t</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">o</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="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c6">p</font>;<br/>
<b>assume </b><a NAME="E2:210_1_2"/><i><font color="Green" title="E108">C1</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">o</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="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c6">p</font> is   <a href="abcmiz_1.html#NM6" title="ABCMIZ_1:NM.6">expression</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">P<sub>1</sub></font>[<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">o</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="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c6">p</font>]</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="210_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:210_1_2_1"/>
(  ex <font color="Olive" title="b1">x</font> being  <a href="abcmiz_1.html#NM3" title="ABCMIZ_1:NM.3">variable</a> st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">o</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="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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>() or  ex <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>() ex <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 <br/>(  <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; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">o</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="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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> ) or  ex <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 <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">o</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="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#K30" title="ABCMIZ_1:func.30">term</a> <font color="Olive" title="b1">a</font> or  ex <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>() ex <font color="Olive" title="b2">q</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 <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">o</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="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="abcmiz_1.html#K31" title="ABCMIZ_1:func.31">term</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">q</font> )
 </span><b>by </b><i><a class="txt" href="abcmiz_1.html#E2:210_1_2"><i><font color="Green" title="E108">C1</font></i></a>, <a class="ref" href="abcmiz_1.html#T53" target="_self" title="ABCMIZ_1:th.53">Th100</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="210_1_2_1_1"><b>suppose </b></a><a NAME="E1:210_1_2_1_1"/>
 ex <font color="Olive" title="b1">x</font> being  <a href="abcmiz_1.html#NM3" title="ABCMIZ_1:NM.3">variable</a> st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">o</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="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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>()
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">o</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="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c6">p</font>]</span><br/><div class="add"><b>hence </b><a NAME="E2:210_1_2_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">o</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="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c6">p</font>]
 <b>by </b><i><a class="txt" href="abcmiz_1.html#E1:210"><i><font color="Green" title="E100">W1</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="210_1_2_1_2"><b>suppose </b></a><a NAME="E1:210_1_2_1_2"/>
 ex <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>() ex <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 <br/>(  <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; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">o</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="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">o</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="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c6">p</font>]</span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c8">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>(), <font color="Maroon" title="c9">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>()<b> such that </b><br/><a NAME="E3:210_1_2_1_2"/><i><font color="Green" title="E109">C3</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">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> <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="c8">c</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">o</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="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c</font> <a href="abcmiz_1.html#K36" title="ABCMIZ_1:func.36">-trm</a> <font color="Maroon" title="c9">q</font> )
 ;<br/><a NAME="E4:210_1_2_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">o</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="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">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="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c9">q</font>
 <b>by </b><i><a class="txt" href="abcmiz_1.html#E3:210_1_2_1_2"><i><font color="Green" title="E109">C3</font></i></a>, <a class="ref" href="abcmiz_1.html#D35" target="_self" title="ABCMIZ_1:def.35">TERM</a></i>;<br/><a NAME="E5:210_1_2_1_2"/><b>then </b><i><font color="Green" title="E110">C4</font></i>: 
<font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font>
 <b>by </b><i><a class="ref" href="trees_4.html#T15" title="TREES_4:th.15">TREES_4:15</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="210_1_2_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">t</font> be   <a href="abcmiz_1.html#NM9" title="ABCMIZ_1:NM.9">quasi-term</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="c10">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="c9">q</font> implies <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c10">t</font>] )</span><br/><a NAME="E1:210_1_2_1_2_1"/>
<font color="Maroon" title="c10">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/><b>hence </b><a NAME="E2:210_1_2_1_2_1"/>
( <font color="Maroon" title="c10">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="c9">q</font> implies <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c10">t</font>] )
 <b>by </b><i><a class="txt" href="abcmiz_1.html#E1:210_1_2"><i><font color="Green" title="E107">Z0</font></i></a>, <a class="txt" href="abcmiz_1.html#E5:210_1_2_1_2"><i><font color="Green" title="E110">C4</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>hence </b><a NAME="E7:210_1_2_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">o</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="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c6">p</font>]
 <b>by </b><i><a class="txt" href="abcmiz_1.html#E3:210_1_2_1_2"><i><font color="Green" title="E109">C3</font></i></a>, <a class="txt" href="abcmiz_1.html#E2:210"><i><font color="Green" title="E101">W2</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="210_1_2_1_3"><b>suppose </b></a><a NAME="E1:210_1_2_1_3"/>
 ex <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 <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">o</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="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#K30" title="ABCMIZ_1:func.30">term</a> <font color="Olive" title="b1">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">o</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="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c6">p</font>]</span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c8">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>()<b> such that </b><br/><a NAME="E3:210_1_2_1_3"/><i><font color="Green" title="E109">C4</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">o</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="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#K30" title="ABCMIZ_1:func.30">term</a> <font color="Maroon" title="c8">a</font>
 ;<br/><a NAME="E4:210_1_2_1_3"/>
(  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_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> <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><span class="p2">(<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></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> &amp; <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><span class="p2">(<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></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <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>() &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><span class="p2">(<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></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="ref" href="abcmiz_1.html#D9" target="_self" title="ABCMIZ_1:def.9">CONSTRSIGN</a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>;<br/><a NAME="E5:210_1_2_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="c5">o</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="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><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="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c8">a</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="abcmiz_1.html#E3:210_1_2_1_3"><i><font color="Green" title="E109">C4</font></i></a>, <a class="ref" href="abcmiz_1.html#D30" target="_self" title="ABCMIZ_1:def.30">TERM1</a></i>;<br/><a NAME="E6:210_1_2_1_3"/><b>then </b><i><font color="Green" title="E110">00</font></i>: 
<font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c8">a</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="trees_4.html#T15" title="TREES_4:th.15">TREES_4:15</a></i>;<br/><a NAME="E7:210_1_2_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="c8">a</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="c8">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c8">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c8">a</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="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a>, <a class="ref" href="msafree3.html#T9" title="MSAFREE3:th.9">MSAFREE3:9</a></i>;<br/><b>hence </b><a NAME="E8:210_1_2_1_3"/>
<font color="Maroon">P<sub>1</sub></font>[<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">o</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="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c6">p</font>]
 <b>by </b><i><a class="txt" href="abcmiz_1.html#E3:210_1_2_1_3"><i><font color="Green" title="E109">C4</font></i></a>, <a class="txt" href="abcmiz_1.html#E3:210"><i><font color="Green" title="E102">W3</font></i></a>, <a class="txt" href="abcmiz_1.html#E6:210_1_2_1_3"><i><font color="Green" title="E110">00</font></i></a>, <a class="txt" href="abcmiz_1.html#E1:210_1_2"><i><font color="Green" title="E107">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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="210_1_2_1_4"><b>suppose </b></a><a NAME="E1:210_1_2_1_4"/>
 ex <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>() ex <font color="Olive" title="b2">q</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 <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">o</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="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="abcmiz_1.html#K31" title="ABCMIZ_1:func.31">term</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">o</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="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c6">p</font>]</span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c8">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>(), <font color="Maroon" title="c9">q</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>()<b> such that </b><br/><a NAME="E3:210_1_2_1_4"/><i><font color="Green" title="E109">C5</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">o</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="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="abcmiz_1.html#K31" title="ABCMIZ_1:func.31">term</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">q</font>
 ;<br/><a NAME="E4:210_1_2_1_4"/>
(  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_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> <span class="p1"><a href="lang1.html#K4" title="LANG1:func.4">&lt;*</a><span class="default"><span class="p2">(<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>,<span class="p2">(<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></span><a href="lang1.html#K4" title="LANG1:func.4">*&gt;</a></span> &amp; <span class="p1"><a href="lang1.html#K4" title="LANG1:func.4">&lt;*</a><span class="default"><span class="p2">(<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>,<span class="p2">(<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></span><a href="lang1.html#K4" title="LANG1:func.4">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <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>() &amp; <span class="p1"><a href="lang1.html#K4" title="LANG1:func.4">&lt;*</a><span class="default"><span class="p2">(<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>,<span class="p2">(<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></span><a href="lang1.html#K4" title="LANG1:func.4">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <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>() &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="lang1.html#K4" title="LANG1:func.4">&lt;*</a><span class="default"><span class="p2">(<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>,<span class="p2">(<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></span><a href="lang1.html#K4" title="LANG1:func.4">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 <b>by </b><i><a class="ref" href="abcmiz_1.html#D9" target="_self" title="ABCMIZ_1:def.9">CONSTRSIGN</a>, <a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>;<br/><a NAME="E5:210_1_2_1_4"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">o</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="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><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="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <span class="p1"><a href="lang1.html#K4" title="LANG1:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">q</font></span><a href="lang1.html#K4" title="LANG1:func.4">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="abcmiz_1.html#E3:210_1_2_1_4"><i><font color="Green" title="E109">C5</font></i></a>, <a class="ref" href="abcmiz_1.html#D31" target="_self" title="ABCMIZ_1:def.31">TERM2</a></i>;<br/><a NAME="E6:210_1_2_1_4"/><b>then </b><i><font color="Green" title="E110">00</font></i>: 
<font color="Maroon" title="c6">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="c8">a</font>,<font color="Maroon" title="c9">q</font></span><a href="lang1.html#K4" title="LANG1:func.4">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="trees_4.html#T15" title="TREES_4:th.15">TREES_4:15</a></i>;<br/><a NAME="E7:210_1_2_1_4"/>
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1"><a href="lang1.html#K4" title="LANG1:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">q</font></span><a href="lang1.html#K4" title="LANG1:func.4">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">q</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> &amp; <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">q</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> &amp; <font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">q</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> &amp; <font color="Maroon" title="c8">a</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> &amp; <font color="Maroon" title="c9">q</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="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a>, <a class="ref" href="finseq_2.html#T147" title="FINSEQ_2:th.147">FINSEQ_2:147</a>, <a class="ref" href="msafree3.html#T9" title="MSAFREE3:th.9">MSAFREE3:9</a></i>;<br/><a NAME="E8:210_1_2_1_4"/><b>then </b>
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c8">a</font>] &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c9">q</font>] )
 <b>by </b><i><a class="txt" href="abcmiz_1.html#E6:210_1_2_1_4"><i><font color="Green" title="E110">00</font></i></a>, <a class="txt" href="abcmiz_1.html#E1:210_1_2"><i><font color="Green" title="E107">Z0</font></i></a></i>;<br/><b>hence </b><a NAME="E9:210_1_2_1_4"/>
<font color="Maroon">P<sub>1</sub></font>[<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">o</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="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c6">p</font>]
 <b>by </b><i><a class="txt" href="abcmiz_1.html#E3:210_1_2_1_4"><i><font color="Green" title="E109">C5</font></i></a>, <a class="txt" href="abcmiz_1.html#E4:210"><i><font color="Green" title="E103">W4</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>end;</b></div>

</div><b>end;</b></div>
<a NAME="E4:210_1"/>
for <font color="Olive" title="b1">t</font> being  <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> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">t</font>]
 
<b>from </b><i><a class="ref" href="msaterm.html#S1" title="MSATERM:sch.1">MSATERM:sch 1</a>(<a class="txt" href="abcmiz_1.html#E2:210_1"><i><font color="Green" title="E105">B1</font></i></a>, <a class="txt" href="abcmiz_1.html#E3:210_1"><i><font color="Green" title="E106">B2</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:210_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>2</sub></font>()]
 <b>by </b><i><a class="txt" href="abcmiz_1.html#E1:210_1"><i><font color="Green" title="E104">B0</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>
