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

<b>let </b><font color="Maroon" title="c1">S1</font>, <font color="Maroon" title="c2">S2</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> non <a href="msualg_1.html#V2" title="MSUALG_1:attr.2">void</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon" title="c3">X</font> be  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V5</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S2</font>;<br/><b>let </b><font color="Maroon" title="c4">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S2</font>;<br/><b>let </b><font color="Maroon" title="c5">g</font> be  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>;<br/>







<b>assume </b><a NAME="E1:55"/><i><font color="Green" title="E31">A1</font></i>: 
<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font> <a href="pua2mss1.html#R3" title="PUA2MSS1:pred.3">form_morphism_between</a> <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c6">h</font> = <font color="Maroon" title="c5">g</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c1">S1</font>,the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c2">S2</font> <b>by </b><i><a class="ref" href="instalg1.html#T10" target="_self" title="INSTALG1:th.10">Th10</a></i>;<br/>
<b>set </b><font color="Maroon" title="c7">A</font> = <span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="instalg1.html#K1" title="INSTALG1:func.1">|</a> <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">B</font> = <span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="instalg1.html#K1" title="INSTALG1:func.1">|</a> <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font> as  <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c1">S1</font> <b>by </b><i><a class="txt" href="instalg1.html#E1:55"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="instalg1.html#T23" target="_self" title="INSTALG1:th.23">Th23</a></i>;<br/>
<b>set </b><font color="Maroon" title="c9">H</font> =  <a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">H'</font> =  <a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font> as   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>,<font color="Maroon" title="c8">B</font> ;<br/>
<b>thus </b><a NAME="E5:55"/>
 <a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a>  <a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c4">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="instalg1.html#K1" title="INSTALG1:func.1">|</a> <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>
 <b>by </b><i><a class="txt" href="instalg1.html#E1:55"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="instalg1.html#D5" target="_self" title="INSTALG1:def.5">Def5</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_3.html#D11" title="MSUALG_3:def.11">MSUALG_3:def 11</a><br/>

<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> ex <font color="Olive" title="b1">t1</font> being  <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon" title="c1">S1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c4">f</font></span>)</span> st <br/>( <font color="Olive" title="b1">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; ( for <font color="Olive" title="b2">t2</font> being  <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon" title="c1">S1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c4">f</font></span>)</span>  st  <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Olive" title="b1">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Olive" title="b2">t2</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Olive" title="b1">t1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Olive" title="b1">t1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">t2</font> holds <br/><font color="Olive" title="b1">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">t2</font> ) );<br/>
<a NAME="E6:55"/><i><font color="Green" title="E32">A2</font></i>: 
 <a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#G3" title="MSUALG_1:aggr.3">MSAlgebra</a>(# <span class="p1">(<span class="default"><a href="msafree.html#K8" title="MSAFREE:func.8">FreeSort</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="msafree.html#K10" title="MSAFREE:func.10">FreeOper</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> #)
 
<b>by </b><i><a class="ref" href="msafree.html#D16" title="MSAFREE:def.16">MSAFREE:def 16</a></i>;<br/>
<a NAME="E7:55"/><i><font color="Green" title="E33">A3</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" title="c1">S1</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"><font color="Maroon" title="c3">X</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c4">f</font></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#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">v</font>,<font color="Olive" title="b1">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">s</font> be   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S1</font>;<br/><b>let </b><font color="Maroon" title="c12">v</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">s</font>;<br/>



<b>reconsider </b><font color="Maroon" title="c13">t1</font> =  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">v</font>,<font color="Maroon" title="c11">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as   <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon" title="c1">S1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <b>by </b><i><a class="ref" href="msaterm.html#T4" title="MSATERM:th.4">MSATERM:4</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c13">t1</font>
;<br/>

<b>thus </b><a NAME="E2:55_1"/>
<font color="Maroon" title="c13">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">v</font>,<font color="Maroon" title="c11">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 ;<br/>

<b>let </b><font color="Maroon" title="c14">t2</font> be   <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon" title="c1">S1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c4">f</font></span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E3:55_1"/><i><font color="Green" title="E34">A4</font></i>: 
 <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c13">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c14">t2</font>
 <b>and </b><br/><a NAME="E4:55_1"/><i><font color="Green" title="E35">A5</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c13">t1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c13">t1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">t2</font>
 ;<br/>

<a NAME="E5:55_1"/><i><font color="Green" title="E36">A6</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c11">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">v</font>,<font color="Maroon" title="c11">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">v</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">s</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="instalg1.html#E1:55"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="instalg1.html#D5" target="_self" title="INSTALG1:def.5">Def5</a></i>;<br/>
<a NAME="E6:55_1"/><i><font color="Green" title="E37">A7</font></i>: 
 <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c13">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">s</font>
 
<b>by </b><i><a class="ref" href="msaterm.html#T14" title="MSATERM:th.14">MSATERM:14</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:55_1_1"/>
<font color="Maroon" title="c14">t2</font> <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#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c1">S1</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ;<br/><a NAME="E2:55_1_1"/><b>then </b>
<font color="Maroon" title="c14">t2</font> is    <a href="msaterm.html#M2" title="MSATERM:mode.2">CompoundTerm</a> of <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c3">X</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c4">f</font>
 <b>by </b><i><a class="ref" href="msaterm.html#D6" title="MSATERM:def.6">MSATERM:def 6</a></i>;<br/><a NAME="E3:55_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">v</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">s</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</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="c12">v</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">s</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">v</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">s</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> is    <a href="msaterm.html#M2" title="MSATERM:mode.2">CompoundTerm</a> of <font color="Maroon" title="c2">S2</font>,<font color="Maroon" title="c3">X</font> )
 <b>by </b><i><a class="txt" href="instalg1.html#E1:55"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="instalg1.html#E3:55_1"><i><font color="Green" title="E34">A4</font></i></a>, <a class="txt" href="instalg1.html#E4:55_1"><i><font color="Green" title="E35">A5</font></i></a>, <a class="txt" href="instalg1.html#E5:55_1"><i><font color="Green" title="E36">A6</font></i></a>, <a class="txt" href="instalg1.html#E6:55_1"><i><font color="Green" title="E37">A7</font></i></a>, <a class="ref" href="instalg1.html#T42" target="_self" title="INSTALG1:th.42">Th42</a>, <a class="ref" href="trees_4.html#T3" title="TREES_4:th.3">TREES_4:3</a></i>;<br/><a NAME="E4:55_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">v</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">s</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c2">S2</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="msaterm.html#D6" title="MSATERM:def.6">MSATERM:def 6</a></i>;<br/><a NAME="E5:55_1_1"/><b>then </b>
( <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S2</font> &amp; <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">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" title="c2">S2</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T129" title="ZFMISC_1:th.129">ZFMISC_1:129</a></i>;<br/><b>hence </b><a NAME="E6:55_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c15">s2</font> being   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S1</font>, <font color="Maroon" title="c16">v2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">s2</font><b> such that </b><br/><a NAME="E9:55_1"/><i><font color="Green" title="E38">A8</font></i>: 
<font color="Maroon" title="c14">t2</font> <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="c16">v2</font>,<font color="Maroon" title="c15">s2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="msaterm.html#T2" title="MSATERM:th.2">MSATERM:2</a></i>;<br/>
<a NAME="E10:55_1"/><i><font color="Green" title="E39">A9</font></i>: 
<font color="Maroon" title="c14">t2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">v2</font>,<font color="Maroon" title="c15">s2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="instalg1.html#E9:55_1"><i><font color="Green" title="E38">A8</font></i></a>, <a class="ref" href="msaterm.html#T5" title="MSATERM:th.5">MSATERM:5</a></i>;<br/>
<a NAME="E11:55_1"/><b>then </b><i><font color="Green" title="E40">A10</font></i>: 
<font color="Maroon" title="c11">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">s2</font>
 
<b>by </b><i><a class="txt" href="instalg1.html#E3:55_1"><i><font color="Green" title="E34">A4</font></i></a>, <a class="txt" href="instalg1.html#E6:55_1"><i><font color="Green" title="E37">A7</font></i></a>, <a class="ref" href="msaterm.html#T14" title="MSATERM:th.14">MSATERM:14</a></i>;<br/>
<a NAME="E12:55_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c11">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">t2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">v2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">s</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="instalg1.html#E1:55"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="instalg1.html#E10:55_1"><i><font color="Green" title="E39">A9</font></i></a>, <a class="ref" href="instalg1.html#D5" target="_self" title="INSTALG1:def.5">Def5</a></i>;<br/>
<a NAME="E13:55_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">v</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">s</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">v2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">s</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="instalg1.html#E4:55_1"><i><font color="Green" title="E35">A5</font></i></a>, <a class="txt" href="instalg1.html#E5:55_1"><i><font color="Green" title="E36">A6</font></i></a>, <a class="txt" href="instalg1.html#E6:55_1"><i><font color="Green" title="E37">A7</font></i></a>, <a class="ref" href="trees_4.html#T4" title="TREES_4:th.4">TREES_4:4</a></i>;<br/>
<b>hence </b><a NAME="E14:55_1"/>
<font color="Maroon" title="c13">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">t2</font>
 <b>by </b><i><a class="txt" href="instalg1.html#E10:55_1"><i><font color="Green" title="E39">A9</font></i></a>, <a class="txt" href="instalg1.html#E11:55_1"><i><font color="Green" title="E40">A10</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:55"/><i><font color="Green" title="E34">A11</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" title="c1">S1</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"><font color="Maroon" title="c3">X</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c4">f</font></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" title="c1">S1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c4">f</font></span>)</span>  st <font color="Olive" title="b3">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">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" title="c1">S1</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="55_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">o</font> be   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S1</font>;<br/><b>let </b><font color="Maroon" title="c12">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="c11">o</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c4">f</font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:55_2"/><i><font color="Green" title="E35">A12</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" title="c1">S1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c4">f</font></span>)</span>  st <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">p</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">t</font>]
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c13">a</font> = <font color="Maroon" title="c12">p</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c11">o</font>,<span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="instalg1.html#T2" target="_self" title="INSTALG1:th.2">Th2</a></i>;<br/>
<b>take </b><font color="Maroon" title="c14">t1</font> = <span class="p1">(<span class="default"><a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> <font color="Maroon" title="c11">o</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="msaterm.html#K6" title="MSATERM:func.6">-tree</a> <font color="Maroon" title="c12">p</font>;<br/>

<b>thus </b><a NAME="E3:55_2"/><i><font color="Green" title="E36">A13</font></i>: 
<font color="Maroon" title="c14">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S1</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="c12">p</font>
 <b>by </b><i><a class="ref" href="msafree.html#D11" title="MSAFREE:def.11">MSAFREE:def 11</a></i>;<br/>

<b>let </b><font color="Maroon" title="c15">t2</font> be   <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon" title="c1">S1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c4">f</font></span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E4:55_2"/><i><font color="Green" title="E37">A14</font></i>: 
 <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c14">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c15">t2</font>
 <b>and </b><br/><a NAME="E5:55_2"/><i><font color="Green" title="E38">A15</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c14">t1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c14">t1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">t2</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c16">q</font> = <span class="p1">(<span class="default"><a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span>)</span> <a href="msualg_3.html#K5" title="MSUALG_3:func.5">#</a> <font color="Maroon" title="c13">a</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">o</font></span>)</span>,<span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon" title="c3">X</font></span>)</span> <b>by </b><i><a class="txt" href="instalg1.html#E1:55"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="instalg1.html#T25" target="_self" title="INSTALG1:th.25">Th25</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">b</font> = <font color="Maroon" title="c16">q</font> as    <a href="msaterm.html#M1" title="MSATERM:mode.1">ArgumentSeq</a> of  <a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">o</font></span>)</span>,<font color="Maroon" title="c3">X</font> <b>by </b><i><a class="ref" href="instalg1.html#T2" target="_self" title="INSTALG1:th.2">Th2</a></i>;<br/>
<a NAME="E8:55_2"/>
 <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c14">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c11">o</font>
 
<b>by </b><i><a class="ref" href="msaterm.html#T20" title="MSATERM:th.20">MSATERM:20</a></i>;<br/>
<a NAME="E9:55_2"/><b>then </b><i><font color="Green" title="E39">A16</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c14">t1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">t1</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"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">o</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S2</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="c16">q</font>
 
<b>by </b><i><a class="txt" href="instalg1.html#E1:55"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="instalg1.html#E3:55_2"><i><font color="Green" title="E36">A13</font></i></a>, <a class="ref" href="instalg1.html#T41" target="_self" title="INSTALG1:th.41">Th41</a></i>;<br/>
<a NAME="E10:55_2"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">o</font></span>)</span>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="msaterm.html#K6" title="MSATERM:func.6">-tree</a> <font color="Maroon" title="c17">b</font></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>  <a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">o</font></span>)</span>,<font color="Maroon" title="c3">X</font> &amp;  <a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">o</font></span>)</span>,<font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">o</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">o</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c2">S2</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="msafree.html#D11" title="MSAFREE:def.11">MSAFREE:def 11</a>, <a class="ref" href="trees_4.html#D4" title="TREES_4:def.4">TREES_4:def 4</a>, <a class="ref" href="zfmisc_1.html#T129" title="ZFMISC_1:th.129">ZFMISC_1:129</a></i>;<br/>
<a NAME="E11:55_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">o</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S2</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="c17">b</font> is    <a href="msaterm.html#M2" title="MSATERM:mode.2">CompoundTerm</a> of <font color="Maroon" title="c2">S2</font>,<font color="Maroon" title="c3">X</font>
 
<b>by </b><i><a class="ref" href="msaterm.html#D6" title="MSATERM:def.6">MSATERM:def 6</a></i>;<br/>
<a NAME="E12:55_2"/><b>then </b>
<font color="Maroon" title="c15">t2</font> is    <a href="msaterm.html#M2" title="MSATERM:mode.2">CompoundTerm</a> of <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c3">X</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c4">f</font>
 
<b>by </b><i><a class="txt" href="instalg1.html#E1:55"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="instalg1.html#E4:55_2"><i><font color="Green" title="E37">A14</font></i></a>, <a class="txt" href="instalg1.html#E5:55_2"><i><font color="Green" title="E38">A15</font></i></a>, <a class="txt" href="instalg1.html#E9:55_2"><i><font color="Green" title="E39">A16</font></i></a>, <a class="ref" href="instalg1.html#T42" target="_self" title="INSTALG1:th.42">Th42</a></i>;<br/>
<a NAME="E13:55_2"/><b>then </b>
<font color="Maroon" title="c15">t2</font> <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#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c1">S1</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="msaterm.html#D6" title="MSATERM:def.6">MSATERM:def 6</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c18">o'</font>, <font color="Maroon" title="c19">s'</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E15:55_2"/><i><font color="Green" title="E40">A17</font></i>: 
( <font color="Maroon" title="c18">o'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c1">S1</font> &amp; <font color="Maroon" title="c19">s'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">t2</font> <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="c18">o'</font>,<font color="Maroon" title="c19">s'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c20">o'</font> = <font color="Maroon" title="c18">o'</font> as   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S1</font> <b>by </b><i><a class="txt" href="instalg1.html#E15:55_2"><i><font color="Green" title="E40">A17</font></i></a></i>;<br/>
<a NAME="E17:55_2"/><i><font color="Green" title="E41">A18</font></i>: 
<font color="Maroon" title="c15">t2</font> <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="c20">o'</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="instalg1.html#E15:55_2"><i><font color="Green" title="E40">A17</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c21">r</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="c20">o'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c4">f</font></span>)</span><b> such that </b><br/><a NAME="E19:55_2"/><i><font color="Green" title="E42">A19</font></i>: 
<font color="Maroon" title="c15">t2</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="c20">o'</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S1</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="c21">r</font>
 <b>by </b><i><a class="ref" href="msaterm.html#T10" title="MSATERM:th.10">MSATERM:10</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c22">c</font> = <font color="Maroon" title="c21">r</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c20">o'</font>,<span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="instalg1.html#T2" target="_self" title="INSTALG1:th.2">Th2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c23">R</font> = <span class="p1">(<span class="default"><a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span>)</span> <a href="msualg_3.html#K5" title="MSUALG_3:func.5">#</a> <font color="Maroon" title="c22">c</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">o'</font></span>)</span>,<span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon" title="c3">X</font></span>)</span> <b>by </b><i><a class="txt" href="instalg1.html#E1:55"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="instalg1.html#T25" target="_self" title="INSTALG1:th.25">Th25</a></i>;<br/>
<a NAME="E22:55_2"/>
 <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c20">o'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c15">t2</font>
 
<b>by </b><i><a class="txt" href="instalg1.html#E17:55_2"><i><font color="Green" title="E41">A18</font></i></a>, <a class="ref" href="msaterm.html#T17" title="MSATERM:th.17">MSATERM:17</a></i>;<br/>
<a NAME="E23:55_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c14">t1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">t1</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"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">o'</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S2</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="c23">R</font>
 
<b>by </b><i><a class="txt" href="instalg1.html#E1:55"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="instalg1.html#E4:55_2"><i><font color="Green" title="E37">A14</font></i></a>, <a class="txt" href="instalg1.html#E5:55_2"><i><font color="Green" title="E38">A15</font></i></a>, <a class="txt" href="instalg1.html#E19:55_2"><i><font color="Green" title="E42">A19</font></i></a>, <a class="ref" href="instalg1.html#T41" target="_self" title="INSTALG1:th.41">Th41</a></i>;<br/>
<a NAME="E24:55_2"/><b>then </b><i><font color="Green" title="E43">A20</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">o'</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">o</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c16">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">R</font> )
 
<b>by </b><i><a class="txt" href="instalg1.html#E9:55_2"><i><font color="Green" title="E39">A16</font></i></a>, <a class="ref" href="trees_4.html#T15" title="TREES_4:th.15">TREES_4:15</a></i>;<br/>
<a NAME="E25:55_2"/><b>then </b>
<font color="Maroon" title="c6">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">o'</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E26:55_2"/><b>then </b><i><font color="Green" title="E44">A21</font></i>: 
<font color="Maroon" title="c11">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">o'</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T25" title="FUNCT_2:th.25">FUNCT_2:25</a></i>;<br/>
<a NAME="E27:55_2"/><b>then </b><i><font color="Green" title="E45">A22</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</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="c11">o</font></span>)</span> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</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="c11">o</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="msaterm.html#T22" title="MSATERM:th.22">MSATERM:22</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c24">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:55_2_1"/><i><font color="Green" title="E46">A23</font></i>: 
<font color="Maroon" title="c24">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c11">o</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c25">w1</font> = <font color="Maroon" title="c12">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font>, <font color="Maroon" title="c26">w2</font> = <font color="Maroon" title="c21">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font> as   <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon" title="c1">S1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <b>by </b><i><a class="txt" href="instalg1.html#E27:55_2"><i><font color="Green" title="E45">A22</font></i></a>, <a class="ref" href="msaterm.html#T22" title="MSATERM:th.22">MSATERM:22</a></i>;<br/><a NAME="E3:55_2_1"/>
<font color="Maroon" title="c25">w1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">p</font>
 <b>by </b><i><a class="txt" href="instalg1.html#E27:55_2"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="instalg1.html#E1:55_2_1"><i><font color="Green" title="E46">A23</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:55_2_1"/><b>then </b><i><font color="Green" title="E47">A24</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c25">w1</font>]
 <b>by </b><i><a class="txt" href="instalg1.html#E1:55_2"><i><font color="Green" title="E35">A12</font></i></a></i>;<br/><a NAME="E5:55_2_1"/><i><font color="Green" title="E48">A25</font></i>: 
(  <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c25">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c11">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">i</font> &amp;  <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c26">w2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c11">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">i</font> )
 <b>by </b><i><a class="txt" href="instalg1.html#E26:55_2"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="instalg1.html#E27:55_2"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="instalg1.html#E1:55_2_1"><i><font color="Green" title="E46">A23</font></i></a>, <a class="ref" href="msaterm.html#T23" title="MSATERM:th.23">MSATERM:23</a></i>;<br/><a NAME="E6:55_2_1"/>
<font color="Maroon" title="c16">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">H'</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c13">a</font>
 ;<br/><a NAME="E7:55_2_1"/><b>then </b>
( <font color="Maroon" title="c16">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c25">w1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">w1</font> &amp; <font color="Maroon" title="c23">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">H'</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c22">c</font> )
 <b>by </b><i><a class="txt" href="instalg1.html#E27:55_2"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="instalg1.html#E1:55_2_1"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="instalg1.html#E5:55_2_1"><i><font color="Green" title="E48">A25</font></i></a>, <a class="ref" href="msualg_3.html#D8" title="MSUALG_3:def.8">MSUALG_3:def 8</a></i>;<br/><a NAME="E8:55_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c25">w1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c25">w1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">w2</font>
 <b>by </b><i><a class="txt" href="instalg1.html#E24:55_2"><i><font color="Green" title="E43">A20</font></i></a>, <a class="txt" href="instalg1.html#E26:55_2"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="instalg1.html#E27:55_2"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="instalg1.html#E1:55_2_1"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="instalg1.html#E5:55_2_1"><i><font color="Green" title="E48">A25</font></i></a>, <a class="ref" href="msualg_3.html#D8" title="MSUALG_3:def.8">MSUALG_3:def 8</a></i>;<br/><b>hence </b><a NAME="E9:55_2_1"/>
<font color="Maroon" title="c12">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font>
 <b>by </b><i><a class="txt" href="instalg1.html#E4:55_2_1"><i><font color="Green" title="E47">A24</font></i></a>, <a class="txt" href="instalg1.html#E5:55_2_1"><i><font color="Green" title="E48">A25</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E29:55_2"/>
<font color="Maroon" title="c14">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">t2</font>
 <b>by </b><i><a class="txt" href="instalg1.html#E3:55_2"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="instalg1.html#E19:55_2"><i><font color="Green" title="E42">A19</font></i></a>, <a class="txt" href="instalg1.html#E26:55_2"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="instalg1.html#E27:55_2"><i><font color="Green" title="E45">A22</font></i></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:55"/><i><font color="Green" title="E35">A26</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" title="c1">S1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c4">f</font></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="instalg1.html#E7:55"><i><font color="Green" title="E33">A3</font></i></a>, <a class="txt" href="instalg1.html#E8:55"><i><font color="Green" title="E34">A11</font></i></a>);<br/></i>
<b>let </b><font color="Maroon" title="c11">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_3.html#D2" title="MSUALG_3:def.2">MSUALG_3:def 2</a><br/><b>let </b><font color="Maroon" title="c12">h</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>;<br/>



<b>assume </b><a NAME="E10:55"/>
<font color="Maroon" title="c11">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c13">s</font> = <font color="Maroon" title="c11">i</font> as   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S1</font> <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>assume </b><a NAME="E12:55"/><i><font color="Green" title="E36">A27</font></i>: 
<span class="p1">(<span class="default"><a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">h</font>
 ;<br/>

<b>then </b><i><font color="Green" title="E37">A28</font></i>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c12">h</font> = 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">H'</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c13">s</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="msafree.html#K8" title="MSAFREE:func.8">FreeSort</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">s</font>
<b>by </b><i><a class="txt" href="instalg1.html#E6:55"><i><font color="Green" title="E32">A2</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="msafree.html#K7" title="MSAFREE:func.7">FreeSort</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c13">s</font>
<b>by </b><i><a class="ref" href="msafree.html#D13" title="MSAFREE:def.13">MSAFREE:def 13</a></i>
;<br/>
<a NAME="E14:55"/><i><font color="Green" title="E38">A29</font></i>: 
 <a href="msafree.html#K7" title="MSAFREE:func.7">FreeSort</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c13">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">S1</font> <a href="msaterm.html#K1" title="MSATERM:func.1">-Terms</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c4">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="msaterm.html#T12" title="MSATERM:th.12">MSATERM:12</a></i>;<br/>
<b>let </b><font color="Maroon" title="c14">x</font>, <font color="Maroon" title="c15">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a><br/>

<b>assume </b><a NAME="E15:55"/><i><font color="Green" title="E39">A30</font></i>: 
( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c12">h</font> &amp; <font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c12">h</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c16">t1</font> = <font color="Maroon" title="c14">x</font>, <font color="Maroon" title="c17">t2</font> = <font color="Maroon" title="c15">y</font> as   <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon" title="c1">S1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <b>by </b><i><a class="txt" href="instalg1.html#E13:55"><i><font color="Green" title="E37">A28</font></i></a>, <a class="txt" href="instalg1.html#E14:55"><i><font color="Green" title="E38">A29</font></i></a></i>;<br/>
<a NAME="E17:55"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c16">t1</font>] &amp;  <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c16">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">s</font> &amp;  <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c17">t2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">s</font> )
 
<b>by </b><i><a class="txt" href="instalg1.html#E9:55"><i><font color="Green" title="E35">A26</font></i></a>, <a class="txt" href="instalg1.html#E13:55"><i><font color="Green" title="E37">A28</font></i></a>, <a class="txt" href="instalg1.html#E15:55"><i><font color="Green" title="E39">A30</font></i></a>, <a class="ref" href="msaterm.html#D5" title="MSATERM:def.5">MSATERM:def 5</a></i>;<br/>
<b>hence </b><a NAME="E18:55"/>
(  not <font color="Maroon" title="c12">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">y</font> or <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">y</font> )
 <b>by </b><i><a class="txt" href="instalg1.html#E12:55"><i><font color="Green" title="E36">A27</font></i></a></i>;<br/>


</div>
