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

<b>let </b><font color="Maroon">S1</font> be  non <a href="struct_0.html#V3">empty</a> non <a href="msualg_1.html#V2">void</a>  <a href="msualg_1.html#L1">ManySortedSign</a> , <font color="Maroon">S2</font> be  non <a href="struct_0.html#V3">empty</a> non <a href="msualg_1.html#V2">void</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon">X</font> be  <a href="relat_1.html#V2">V5</a>  <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S2</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S2</font>;<br/><b>let </b><font color="Maroon">g</font> be  <a href="funct_1.html#V2">one-to-one</a> <a href="funct_1.html#NM1">Function</a>;<br/>







<b>assume </b><a NAME="E1:55"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">f</font>,<font color="Maroon">g</font> <a href="pua2mss1.html#R3">form_morphism_between</a> <font color="Maroon">S1</font>,<font color="Maroon">S2</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">h</font> = <font color="Maroon">g</font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S1</font>,the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S2</font> <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">A</font> = <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="instalg1.html#K1" target="_self">|</a> <font color="Maroon">S1</font>,<font color="Maroon">f</font>,<font color="Maroon">g</font>;<br/>
<b>reconsider </b><font color="Maroon">B</font> = <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="instalg1.html#K1" target="_self">|</a> <font color="Maroon">S1</font>,<font color="Maroon">f</font>,<font color="Maroon">g</font> as  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S1</font> <b>by </b><i>, </i>;<br/>
<b>set </b><font color="Maroon">H</font> =  <a href="instalg1.html#K3" target="_self">hom</a> <font color="Maroon">f</font>,<font color="Maroon">g</font>,<font color="Maroon">X</font>,<font color="Maroon">S1</font>,<font color="Maroon">S2</font>;<br/>
<b>reconsider </b><font color="Maroon">H'</font> =  <a href="instalg1.html#K3" target="_self">hom</a> <font color="Maroon">f</font>,<font color="Maroon">g</font>,<font color="Maroon">X</font>,<font color="Maroon">S1</font>,<font color="Maroon">S2</font> as   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">B</font> ;<br/>
<b>thus </b><a NAME="E5:55"/>
 <a href="instalg1.html#K3" target="_self">hom</a> <font color="Maroon">f</font>,<font color="Maroon">g</font>,<font color="Maroon">X</font>,<font color="Maroon">S1</font>,<font color="Maroon">S2</font> <a href="msualg_3.html#R1">is_homomorphism</a>  <a href="msafree.html#K11">FreeMSA</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="instalg1.html#K1" target="_self">|</a> <font color="Maroon">S1</font>,<font color="Maroon">f</font>,<font color="Maroon">g</font>
 <b>by </b><i>, </i>; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_3.html#D11">MSUALG_3:def 11</a><br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">t1</font> being  <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S1</font>,<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span> st <br/>( <font color="Olive">t1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; ( for <font color="Olive">t2</font> being  <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S1</font>,<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span>  st  <a href="msaterm.html#K7">the_sort_of</a> <font color="Olive">t1</font> <a href="hidden.html#R1">=</a>  <a href="msaterm.html#K7">the_sort_of</a> <font color="Olive">t2</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="instalg1.html#K3" target="_self">hom</a> <font color="Maroon">f</font>,<font color="Maroon">g</font>,<font color="Maroon">X</font>,<font color="Maroon">S1</font>,<font color="Maroon">S2</font></span>)</span> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="msaterm.html#K7">the_sort_of</a> <font color="Olive">t1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">t1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="instalg1.html#K3" target="_self">hom</a> <font color="Maroon">f</font>,<font color="Maroon">g</font>,<font color="Maroon">X</font>,<font color="Maroon">S1</font>,<font color="Maroon">S2</font></span>)</span> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="msaterm.html#K7">the_sort_of</a> <font color="Olive">t1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">t2</font> holds <br/><font color="Olive">t1</font> <a href="hidden.html#R1">=</a> <font color="Olive">t2</font> ) );<br/>
<a NAME="E6:55"/><i><font color="Green">E41</font></i>: 
 <a href="msafree.html#K11">FreeMSA</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#G3">MSAlgebra</a>(# <span class="p1">(<span class="default"><a href="msafree.html#K8">FreeSort</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="msafree.html#K10">FreeOper</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span></span>)</span> #)
 
<b>by </b><i><a class="ref" href="msafree.html#D16">MSAFREE:def 16</a></i>;<br/>
<a NAME="E7:55"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">s</font> being  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S1</font><br/> for <font color="Olive">v</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font> holds  <font color="Maroon">S<sub>1</sub></font>[ <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">v</font>,<font color="Olive">s</font></span><a href="tarski.html#K4">]</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">s</font> be   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S1</font>;<br/><b>let </b><font color="Maroon">v</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>;<br/>



<b>reconsider </b><font color="Maroon">t1</font> =  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span> as   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S1</font>,<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span> <b>by </b><i><a class="ref" href="msaterm.html#T4">MSATERM:4</a></i>;<br/>
<b>take </b>
<font color="Maroon">t1</font>
;<br/>

<b>thus </b><a NAME="E2:55_1"/>
<font color="Maroon">t1</font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span>
 ;<br/>

<b>let </b><font color="Maroon">t2</font> be   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S1</font>,<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E3:55_1"/><i><font color="Green">E53</font></i>: 
 <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">t1</font> <a href="hidden.html#R1">=</a>  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">t2</font>
 <b>and </b><br/><a NAME="E4:55_1"/><i><font color="Green">E56</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="instalg1.html#K3" target="_self">hom</a> <font color="Maroon">f</font>,<font color="Maroon">g</font>,<font color="Maroon">X</font>,<font color="Maroon">S1</font>,<font color="Maroon">S2</font></span>)</span> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">t1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">t1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="instalg1.html#K3" target="_self">hom</a> <font color="Maroon">f</font>,<font color="Maroon">g</font>,<font color="Maroon">X</font>,<font color="Maroon">S1</font>,<font color="Maroon">S2</font></span>)</span> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">t1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">t2</font>
 ;<br/>

<a NAME="E5:55_1"/><i><font color="Green">E57</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="instalg1.html#K3" target="_self">hom</a> <font color="Maroon">f</font>,<font color="Maroon">g</font>,<font color="Maroon">X</font>,<font color="Maroon">S1</font>,<font color="Maroon">S2</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="trees_4.html#K1">root-tree</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">v</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:55_1"/><i><font color="Green">E81</font></i>: 
 <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">t1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="msaterm.html#T14">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">t2</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S1</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S1</font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><a NAME="E2:55_1_1"/><b>then </b>
<font color="Maroon">t2</font> is    <a href="msaterm.html#M2">CompoundTerm</a> of <font color="Maroon">S1</font>,<font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="msaterm.html#D6">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">root-tree</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">v</font>,<span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font></span>)</span></span><a href="tarski.html#K4">]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">v</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font></span>)</span></span><a href="tarski.html#K4">]</a></span> &amp;  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">v</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font></span>)</span></span><a href="tarski.html#K4">]</a></span> is    <a href="msaterm.html#M2">CompoundTerm</a> of <font color="Maroon">S2</font>,<font color="Maroon">X</font> )
 <b>by </b><i>, , , , <a class="txt" href="instalg1.html#E1:55"><i><font color="Green">E37</font></i></a>, , <a class="ref" href="trees_4.html#T3">TREES_4:3</a></i>;<br/><a NAME="E4:55_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">v</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S2</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S2</font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="msaterm.html#D6">MSATERM:def 6</a></i>;<br/><a NAME="E5:55_1_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S2</font> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S2</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T129">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">s2</font> being   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S1</font>, <font color="Maroon">v2</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s2</font><b> such that </b><br/><a NAME="E9:55_1"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">t2</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">v2</font>,<font color="Maroon">s2</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="msaterm.html#T2">MSATERM:2</a></i>;<br/>
<a NAME="E10:55_1"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">t2</font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">v2</font>,<font color="Maroon">s2</font></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i>, <a class="ref" href="msaterm.html#T5">MSATERM:5</a></i>;<br/>
<a NAME="E11:55_1"/><b>then </b><i><font color="Green">E118</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font>
 
<b>by </b><i>, <a class="txt" href="instalg1.html#E1:55"><i><font color="Green">E37</font></i></a>, <a class="ref" href="msaterm.html#T14">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" target="_self">hom</a> <font color="Maroon">f</font>,<font color="Maroon">g</font>,<font color="Maroon">X</font>,<font color="Maroon">S1</font>,<font color="Maroon">S2</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">t2</font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">v2</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E13:55_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">v</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">v2</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i>, , <a class="txt" href="instalg1.html#E1:55"><i><font color="Green">E37</font></i></a>, <a class="ref" href="trees_4.html#T4">TREES_4:4</a></i>;<br/>
<b>hence </b><a NAME="E14:55_1"/>
<font color="Maroon">t1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t2</font>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:55"/><i><font color="Green">E120</font></i>: 
for <font color="Olive">o</font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S1</font><br/> for <font color="Olive">p</font> being   <a href="msaterm.html#M1">ArgumentSeq</a> of  <a href="msaterm.html#K2">Sym</a> <font color="Olive">o</font>,<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span>  st ( for <font color="Olive">t</font> being  <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S1</font>,<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span>  st <font color="Olive">t</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Olive">p</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">t</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S1</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Olive">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">o</font> be   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S1</font>;<br/><b>let </b><font color="Maroon">p</font> be    <a href="msaterm.html#M1">ArgumentSeq</a> of  <a href="msaterm.html#K2">Sym</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:55_2"/><i><font color="Green">E121</font></i>: 
for <font color="Olive">t</font> being  <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S1</font>,<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span>  st <font color="Olive">t</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">t</font>]
 ;<br/>

<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">p</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span></span>)</span> <b>by </b><i/>;<br/>
<b>take </b><font color="Maroon">t1</font> = <span class="p1">(<span class="default"><a href="msaterm.html#K2">Sym</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="msaterm.html#K6">-tree</a> <font color="Maroon">p</font>;<br/>

<b>thus </b><a NAME="E3:55_2"/><i><font color="Green">E141</font></i>: 
<font color="Maroon">t1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S1</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="msafree.html#D11">MSAFREE:def 11</a></i>;<br/>

<b>let </b><font color="Maroon">t2</font> be   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S1</font>,<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E4:55_2"/><i><font color="Green">E142</font></i>: 
 <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">t1</font> <a href="hidden.html#R1">=</a>  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">t2</font>
 <b>and </b><br/><a NAME="E5:55_2"/><i><font color="Green">E143</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="instalg1.html#K3" target="_self">hom</a> <font color="Maroon">f</font>,<font color="Maroon">g</font>,<font color="Maroon">X</font>,<font color="Maroon">S1</font>,<font color="Maroon">S2</font></span>)</span> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">t1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">t1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="instalg1.html#K3" target="_self">hom</a> <font color="Maroon">f</font>,<font color="Maroon">g</font>,<font color="Maroon">X</font>,<font color="Maroon">S1</font>,<font color="Maroon">S2</font></span>)</span> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">t1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">t2</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">q</font> = <span class="p1">(<span class="default"><a href="instalg1.html#K3" target="_self">hom</a> <font color="Maroon">f</font>,<font color="Maroon">g</font>,<font color="Maroon">X</font>,<font color="Maroon">S1</font>,<font color="Maroon">S2</font></span>)</span> <a href="msualg_3.html#K5">#</a> <font color="Maroon">a</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font></span>)</span>,<span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">q</font> as    <a href="msaterm.html#M1">ArgumentSeq</a> of  <a href="msaterm.html#K2">Sym</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font></span>)</span>,<font color="Maroon">X</font> <b>by </b><i/>;<br/>
<a NAME="E8:55_2"/>
 <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">t1</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font>
 
<b>by </b><i><a class="ref" href="msaterm.html#T20">MSATERM:20</a></i>;<br/>
<a NAME="E9:55_2"/><b>then </b><i><font color="Green">E144</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="instalg1.html#K3" target="_self">hom</a> <font color="Maroon">f</font>,<font color="Maroon">g</font>,<font color="Maroon">X</font>,<font color="Maroon">S1</font>,<font color="Maroon">S2</font></span>)</span> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">t1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">t1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S2</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">q</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E10:55_2"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msaterm.html#K2">Sym</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font></span>)</span>,<font color="Maroon">X</font></span>)</span> <a href="msaterm.html#K6">-tree</a> <font color="Maroon">b</font></span>)</span> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="msaterm.html#K2">Sym</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font></span>)</span>,<font color="Maroon">X</font> &amp;  <a href="msaterm.html#K2">Sym</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font></span>)</span>,<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S2</font></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S2</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S2</font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="msafree.html#D11">MSAFREE:def 11</a>, <a class="ref" href="trees_4.html#D4">TREES_4:def 4</a>, <a class="ref" href="zfmisc_1.html#T129">ZFMISC_1:129</a></i>;<br/>
<a NAME="E11:55_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S2</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">b</font> is    <a href="msaterm.html#M2">CompoundTerm</a> of <font color="Maroon">S2</font>,<font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="msaterm.html#D6">MSATERM:def 6</a></i>;<br/>
<a NAME="E12:55_2"/><b>then </b>
<font color="Maroon">t2</font> is    <a href="msaterm.html#M2">CompoundTerm</a> of <font color="Maroon">S1</font>,<font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="txt" href="instalg1.html#E1:55"><i><font color="Green">E37</font></i></a>, , , </i>;<br/>
<a NAME="E13:55_2"/><b>then </b>
<font color="Maroon">t2</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S1</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S1</font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="msaterm.html#D6">MSATERM:def 6</a></i>;<br/>
<b>then consider </b><font color="Maroon">o'</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">s'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E15:55_2"/><i><font color="Green">E147</font></i>: 
( <font color="Maroon">o'</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S1</font> &amp; <font color="Maroon">s'</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S1</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">t2</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o'</font>,<font color="Maroon">s'</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">o'</font> = <font color="Maroon">o'</font> as   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S1</font> <b>by </b><i/>;<br/>
<a NAME="E17:55_2"/><i><font color="Green">E148</font></i>: 
<font color="Maroon">t2</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o'</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S1</font></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">r</font> being    <a href="msaterm.html#M1">ArgumentSeq</a> of  <a href="msaterm.html#K2">Sym</a> <font color="Maroon">o'</font>,<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span><b> such that </b><br/><a NAME="E19:55_2"/><i><font color="Green">E149</font></i>: 
<font color="Maroon">t2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o'</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S1</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="msaterm.html#T10">MSATERM:10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c</font> = <font color="Maroon">r</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o'</font>,<span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span></span>)</span> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">R</font> = <span class="p1">(<span class="default"><a href="instalg1.html#K3" target="_self">hom</a> <font color="Maroon">f</font>,<font color="Maroon">g</font>,<font color="Maroon">X</font>,<font color="Maroon">S1</font>,<font color="Maroon">S2</font></span>)</span> <a href="msualg_3.html#K5">#</a> <font color="Maroon">c</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o'</font></span>)</span>,<span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i>, </i>;<br/>
<a NAME="E22:55_2"/>
 <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o'</font> <a href="hidden.html#R1">=</a>  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">t2</font>
 
<b>by </b><i><a class="txt" href="instalg1.html#E6:55"><i><font color="Green">E41</font></i></a>, <a class="ref" href="msaterm.html#T17">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" target="_self">hom</a> <font color="Maroon">f</font>,<font color="Maroon">g</font>,<font color="Maroon">X</font>,<font color="Maroon">S1</font>,<font color="Maroon">S2</font></span>)</span> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">t1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">t1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o'</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S2</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">R</font>
 
<b>by </b><i>, <a class="txt" href="instalg1.html#E1:55"><i><font color="Green">E37</font></i></a>, , <a class="ref" href="instalg1.html#T3" target="_self">Th3</a>, </i>;<br/>
<a NAME="E24:55_2"/><b>then </b><i><font color="Green">E152</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o'</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S2</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> )
 
<b>by </b><i>, <a class="ref" href="trees_4.html#T15">TREES_4:15</a></i>;<br/>
<a NAME="E25:55_2"/><b>then </b>
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o'</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E26:55_2"/><b>then </b><i><font color="Green">E153</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o'</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T25">FUNCT_2:25</a></i>;<br/>
<a NAME="E27:55_2"/><b>then </b><i><font color="Green">E154</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="msaterm.html#T22">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">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:55_2_1"/><i><font color="Green">E155</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">w1</font> = <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>, <font color="Maroon">w2</font> = <font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S1</font>,<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span> <b>by </b><i>, <a class="ref" href="msaterm.html#T22">MSATERM:22</a></i>;<br/><a NAME="E3:55_2_1"/>
<font color="Maroon">w1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:55_2_1"/><b>then </b><i><font color="Green">E158</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">w1</font>]
 <b>by </b><i/>;<br/><a NAME="E5:55_2_1"/><i><font color="Green">E159</font></i>: 
(  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">w1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> &amp;  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">w2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> )
 <b>by </b><i><a class="ref" href="instalg1.html#T4" target="_self">Th4</a>, , , <a class="ref" href="msaterm.html#T23">MSATERM:23</a></i>;<br/><a NAME="E6:55_2_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H'</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">a</font>
 ;<br/><a NAME="E7:55_2_1"/><b>then </b>
( <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="instalg1.html#K3" target="_self">hom</a> <font color="Maroon">f</font>,<font color="Maroon">g</font>,<font color="Maroon">X</font>,<font color="Maroon">S1</font>,<font color="Maroon">S2</font></span>)</span> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">w1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">w1</font> &amp; <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H'</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">c</font> )
 <b>by </b><i>, , , <a class="ref" href="msualg_3.html#D8">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" target="_self">hom</a> <font color="Maroon">f</font>,<font color="Maroon">g</font>,<font color="Maroon">X</font>,<font color="Maroon">S1</font>,<font color="Maroon">S2</font></span>)</span> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">w1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">w1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="instalg1.html#K3" target="_self">hom</a> <font color="Maroon">f</font>,<font color="Maroon">g</font>,<font color="Maroon">X</font>,<font color="Maroon">S1</font>,<font color="Maroon">S2</font></span>)</span> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">w1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">w2</font>
 <b>by </b><i>, <a class="ref" href="instalg1.html#T4" target="_self">Th4</a>, , , , <a class="ref" href="msualg_3.html#D8">MSUALG_3:def 8</a></i>;<br/><b>hence </b><a NAME="E9:55_2_1"/>
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="instalg1.html#T6" target="_self">Th6</a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E29:55_2"/>
<font color="Maroon">t1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t2</font>
 <b>by </b><i>, <a class="ref" href="instalg1.html#T3" target="_self">Th3</a>, <a class="ref" href="instalg1.html#T4" target="_self">Th4</a>, , <a class="ref" href="finseq_1.html#T17">FINSEQ_1:17</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:55"/><i><font color="Green">E160</font></i>: 
for <font color="Olive">t</font> being  <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S1</font>,<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">t</font>]
 
<b>from </b><i><a class="ref" href="msaterm.html#S1">MSATERM:sch 1</a>(, );<br/></i>
<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_3.html#D2">MSUALG_3:def 2</a><br/><b>let </b><font color="Maroon">h</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>



<b>assume </b><a NAME="E10:55"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="instalg1.html#K3" target="_self">hom</a> <font color="Maroon">f</font>,<font color="Maroon">g</font>,<font color="Maroon">X</font>,<font color="Maroon">S1</font>,<font color="Maroon">S2</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">s</font> = <font color="Maroon">i</font> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S1</font> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>assume </b><a NAME="E12:55"/><i><font color="Green">E161</font></i>: 
<span class="p1">(<span class="default"><a href="instalg1.html#K3" target="_self">hom</a> <font color="Maroon">f</font>,<font color="Maroon">g</font>,<font color="Maroon">X</font>,<font color="Maroon">S1</font>,<font color="Maroon">S2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font>
 ;<br/>

<b>then </b><i><font color="Green">E162</font></i>:  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> = 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">H'</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="msafree.html#K8">FreeSort</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
<b>by </b><i><a class="ref" href="instalg1.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="msafree.html#K7">FreeSort</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">s</font>
<b>by </b><i><a class="ref" href="msafree.html#D13">MSAFREE:def 13</a></i>
;<br/>
<a NAME="E14:55"/><i><font color="Green">E163</font></i>: 
 <a href="msafree.html#K7">FreeSort</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">s</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S1</font> <a href="msaterm.html#K1">-Terms</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="msaterm.html#T12">MSATERM:12</a></i>;<br/>
<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a><br/>

<b>assume </b><a NAME="E15:55"/><i><font color="Green">E164</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">t1</font> = <font color="Maroon">x</font>, <font color="Maroon">t2</font> = <font color="Maroon">y</font> as   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S1</font>,<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span> <b>by </b><i><a class="txt" href="instalg1.html#E1:55"><i><font color="Green">E37</font></i></a>, </i>;<br/>
<a NAME="E17:55"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">t1</font>] &amp;  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">t1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp;  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">t2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> )
 
<b>by </b><i>, <a class="txt" href="instalg1.html#E1:55"><i><font color="Green">E37</font></i></a>, , <a class="ref" href="msaterm.html#D5">MSATERM:def 5</a></i>;<br/>
<b>hence </b><a NAME="E18:55"/>
(  not <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 <b>by </b><i/>;<br/>


</div>
