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

<b>reconsider </b><font color="Maroon">A</font> =  <a href="msafree.html#K13">FreeGen</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">MSA</font> as  <a href="msafree.html#V1">free</a>  <a href="msafree.html#M1">GeneratorSet</a> of  <a href="msafree2.html#K6" target="_self">FreeEnv</a> <font color="Maroon">MSA</font> <b>by </b><i><a class="ref" href="msafree.html#T17">MSAFREE:17</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">s</font> being  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font> ex <font color="Olive">f</font> being  <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">MSA</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font> st <br/>( <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; ( for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font> &amp; <font color="Olive">y</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="Olive">x</font>,<font color="Olive">s</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">MSA</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font> holds <br/><font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> ) );<br/>
<a NAME="E2:19_1_1"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> holds <br/> ex <font color="Olive">j</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>,<font color="Olive">j</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:19_1_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 ;<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">S</font> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></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">a<sub>2</sub></font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span>;<br/>
<a NAME="E3:19_1_1_1"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> holds <br/> ex <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">u</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">MSA</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">e</font>,<font color="Olive">u</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:19_1_1_1_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 ;<br/>

<a NAME="E2:19_1_1_1_1"/><b>then </b>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="msafree.html#K12">FreeGen</a> <font color="Maroon">s</font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">MSA</font>
 
<b>by </b><i><a class="ref" href="msafree.html#D18">MSAFREE:def 18</a></i>;<br/>
<b>hence </b><a NAME="E3:19_1_1_1_1"/>
 ex <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">u</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">MSA</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> &amp; <font color="Maroon">e</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="Olive">u</font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="msafree.html#D17">MSAFREE:def 17</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">j</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:19_1_1_1"/><i><font color="Green">E33</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">j</font> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">MSA</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> &amp; ( for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">e</font>,<font color="Maroon">j</font> <a href="funct_1.html#K1">.</a> <font color="Olive">e</font>] ) )
 <b>from </b><i><a class="ref" href="wellord2.html#S1">WELLORD2:sch 1</a>();<br/></i>
<b>take </b>
<font color="Maroon">j</font>
;<br/>

<b>take </b>
<font color="Maroon">s</font>
;<br/>

<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">j</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">MSA</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>thus </b><a NAME="E7:19_1_1_1"/>
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> )
 ;<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> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E8:19_1_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E9:19_1_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">y</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">x</font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E10:19_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">MSA</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 ;<br/>

<a NAME="E11:19_1_1_1"/>
<font color="Maroon">y</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"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E12:19_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">s</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">j</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i>, <a class="ref" href="trees_4.html#T4">TREES_4:4</a></i>;<br/>
<b>hence </b><a NAME="E13:19_1_1_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E4:19_1_1"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>]
 <b>from </b><i><a class="ref" href="pboole.html#S3">PBOOLE:sch 3</a>();<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:19_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 ;<br/><a NAME="E2:19_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E3:19_1_1_2"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>]
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:19_1_1_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> is   <a href="funct_1.html#NM1">Function</a>
 ;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="pboole.html#NM2">ManySortedFunction</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="funcop_1.html#D6">FUNCOP_1:def 6</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:19_1_1_3"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 ;<br/><a NAME="E2:19_1_1_3"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>]
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:19_1_1_3"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">MSA</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as    <a href="pboole.html#M3">ManySortedFunction</a> of <font color="Maroon">A</font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">MSA</font> <b>by </b><i><a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/>
<b>consider </b><font color="Maroon">IT</font> being   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6" target="_self">FreeEnv</a> <font color="Maroon">MSA</font></span>)</span>,<font color="Maroon">MSA</font><b> such that </b><br/><a NAME="E10:19_1_1"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">IT</font> <a href="msualg_3.html#R1">is_homomorphism</a>  <a href="msafree2.html#K6" target="_self">FreeEnv</a> <font color="Maroon">MSA</font>,<font color="Maroon">MSA</font> &amp; <font color="Maroon">IT</font> <a href="msafree.html#K1">||</a> <font color="Maroon">A</font> <a href="pboole.html#R6">=</a> <font color="Maroon">f</font> )
 <b>by </b><i><a class="ref" href="msafree.html#D5">MSAFREE:def 5</a></i>;<br/>
<b>take </b>
<font color="Maroon">IT</font>
;<br/>

<b>thus </b><a NAME="E11:19_1_1"/>
<font color="Maroon">IT</font> <a href="msualg_3.html#R1">is_homomorphism</a>  <a href="msafree2.html#K6" target="_self">FreeEnv</a> <font color="Maroon">MSA</font>,<font color="Maroon">MSA</font>
 <b>by </b><i/>;<br/>

<b>let </b><font color="Maroon">s</font> be   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font>;<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> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E12:19_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="msafree.html#K7">FreeSort</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">MSA</font>,<font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E13:19_1_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">y</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">x</font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E14:19_1_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">MSA</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 ;<br/>

<a NAME="E15:19_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="msafree.html#K12">FreeGen</a> <font color="Maroon">s</font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">MSA</font>
 
<b>by </b><i>, , <a class="ref" href="msafree.html#D17">MSAFREE:def 17</a></i>;<br/>
<a NAME="E16:19_1_1"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="msafree.html#D18">MSAFREE:def 18</a></i>;<br/>
<b>consider </b><font color="Maroon">t</font> being   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font>, <font color="Maroon">g</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">MSA</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font><b> such that </b><br/><a NAME="E18:19_1_1"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">g</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">f</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font> &amp; <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> )
 <b>and </b><br/><a NAME="E19:19_1_1"/><i><font color="Green">E70</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> &amp; <font color="Olive">y</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="Olive">x</font>,<font color="Maroon">t</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">MSA</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> holds <br/><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font>
 <b>by </b><i/>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">IT</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="msafree.html#D1">MSAFREE:def 1</a></i>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i>, , , , </i>
;<br/>


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