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

<b>let </b><font color="Maroon">IT1</font> be   <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>, <font color="Maroon">IT2</font> be   <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>;<br/>

<b>reconsider </b><font color="Maroon">IT1'</font> = <font color="Maroon">IT1</font>, <font color="Maroon">IT2'</font> = <font color="Maroon">IT2</font> as   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">MSA</font></span>)</span>,<font color="Maroon">MSA</font> ;<br/>
<b>assume </b><a NAME="E2:19_1_2"/>
<font color="Maroon">IT1</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>
 ;<br/>

<a NAME="E3:19_1_2"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">IT1'</font> <a href="msualg_3.html#R1">is_homomorphism</a>  <a href="msafree.html#K11">FreeMSA</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">MSA</font>,<font color="Maroon">MSA</font>
 
;<br/>
<b>assume </b><a NAME="E4:19_1_2"/><i><font color="Green">E76</font></i>: 
for <font color="Olive">s</font> being  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font><br/> 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>  <a href="msafree.html#K7">FreeSort</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">MSA</font>,<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/><span class="p1">(<span class="default"><font color="Maroon">IT1</font> <a href="msualg_3.html#K1">.</a> <font color="Olive">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font>
 ;<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> ,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>3</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">a<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span>;<br/>
<a NAME="E5:19_1_2"/><i><font color="Green">E77</font></i>: 
for <font color="Olive">s</font> being  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font><br/> 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>  <a href="msafree.html#K12">FreeGen</a> <font color="Olive">s</font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">MSA</font> holds <br/>( <span class="p1">(<span class="default"><font color="Maroon">IT1'</font> <a href="msualg_3.html#K1">.</a> <font color="Olive">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">s</font>,<font color="Olive">x</font>,<font color="Olive">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_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">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><a NAME="E1:19_1_2_1"/><i><font color="Green">E78</font></i>: 
<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>
 ;<br/>

<a NAME="E2:19_1_2_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="msafree.html#K8">FreeSort</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">MSA</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 
;<br/>
<a NAME="E3:19_1_2_1"/><b>then </b><i><font color="Green">E79</font></i>: 
<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>by </b><i><a class="ref" href="msafree.html#D13">MSAFREE:def 13</a></i>;<br/>
<b>consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:19_1_2_1"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">a</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>
 <b>and </b><br/><a NAME="E6:19_1_2_1"/><i><font color="Green">E82</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">a</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/>
<b>thus </b><a NAME="E7:19_1_2_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">IT1'</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> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> implies <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>by </b><i>, , , </i>;<br/>

<b>assume </b><a NAME="E8:19_1_2_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"><font color="Maroon">x</font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span>
 ;<br/>

<a NAME="E9:19_1_2_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"><font color="Maroon">a</font>,<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/>
<a NAME="E10:19_1_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E11:19_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">IT1'</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> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i>, , , </i>;<br/>


</div><b>end;</b></div>
<b>assume </b><a NAME="E6:19_1_2"/>
<font color="Maroon">IT2</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>
 ;<br/>

<a NAME="E7:19_1_2"/><b>then </b><i><font color="Green">E83</font></i>: 
<font color="Maroon">IT2'</font> <a href="msualg_3.html#R1">is_homomorphism</a>  <a href="msafree.html#K11">FreeMSA</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">MSA</font>,<font color="Maroon">MSA</font>
 
;<br/>
<b>assume </b><a NAME="E8:19_1_2"/><i><font color="Green">E84</font></i>: 
for <font color="Olive">s</font> being  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font><br/> 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>  <a href="msafree.html#K7">FreeSort</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">MSA</font>,<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/><span class="p1">(<span class="default"><font color="Maroon">IT2</font> <a href="msualg_3.html#K1">.</a> <font color="Olive">s</font></span>)</span> <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="E9:19_1_2"/><i><font color="Green">E85</font></i>: 
for <font color="Olive">s</font> being  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font><br/> 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>  <a href="msafree.html#K12">FreeGen</a> <font color="Olive">s</font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">MSA</font> holds <br/>( <span class="p1">(<span class="default"><font color="Maroon">IT2'</font> <a href="msualg_3.html#K1">.</a> <font color="Olive">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">s</font>,<font color="Olive">x</font>,<font color="Olive">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_2"><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">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><a NAME="E1:19_1_2_2"/><i><font color="Green">E86</font></i>: 
<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>
 ;<br/>

<a NAME="E2:19_1_2_2"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="msafree.html#K8">FreeSort</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">MSA</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 
;<br/>
<a NAME="E3:19_1_2_2"/><b>then </b><i><font color="Green">E87</font></i>: 
<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>by </b><i><a class="ref" href="msafree.html#D13">MSAFREE:def 13</a></i>;<br/>
<b>consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:19_1_2_2"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">a</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>
 <b>and </b><br/><a NAME="E6:19_1_2_2"/><i><font color="Green">E89</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">a</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/>
<b>thus </b><a NAME="E7:19_1_2_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">IT2'</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> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> implies <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>by </b><i>, , , </i>;<br/>

<b>assume </b><a NAME="E8:19_1_2_2"/>
<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>
 ;<br/>

<a NAME="E9:19_1_2_2"/><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"><font color="Maroon">a</font>,<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/>
<a NAME="E10:19_1_2_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E11:19_1_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon">IT2'</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> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i>, , , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:19_1_2"/>
<font color="Maroon">IT1'</font> <a href="pboole.html#R6">=</a> <font color="Maroon">IT2'</font>
 
<b>from </b><i><a class="ref" href="msafree1.html#S3">MSAFREE1:sch 3</a>(, , , );<br/></i>
<b>hence </b><a NAME="E11:19_1_2"/>
<font color="Maroon">IT1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">IT2</font>
 ;<br/>


</div>
