<?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#NM1">Function</a>;<br/>







<b>assume </b><a NAME="E1:54"/><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>let </b><font color="Maroon">t</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/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:54_1"/>
<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">t</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> is    <a href="msaterm.html#M2">CompoundTerm</a> of <font color="Maroon">S2</font>,<font color="Maroon">X</font>
 ;<br/><b>then reconsider </b><font color="Maroon">w</font> = <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">t</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> as    <a href="msaterm.html#M2">CompoundTerm</a> of <font color="Maroon">S2</font>,<font color="Maroon">X</font> ;<br/><a NAME="E3:54_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">w</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">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/><b>assume </b><a NAME="E4:54_1"/>
<font color="Maroon">t</font> is  not    <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>
 ;<br/><a NAME="E5:54_1"/><b>then </b>
not <font color="Maroon">t</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">s</font> being   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S1</font>, <font color="Maroon">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="Maroon">s</font><b> such that </b><br/><a NAME="E7:54_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">t</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">v</font>,<font color="Maroon">s</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="E8:54_1"/>
<font color="Maroon">t</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>
 <b>by </b><i>, <a class="ref" href="msaterm.html#T5">MSATERM:5</a></i>;<br/><a NAME="E9:54_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">t</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>,<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="msaterm.html#K7">the_sort_of</a> <font color="Maroon">t</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/><a NAME="E10:54_1"/><b>then </b>
<font color="Maroon">w</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">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><a class="ref" href="trees_4.html#T3">TREES_4:3</a></i>;<br/><a NAME="E11:54_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="instalg1.html#T2" target="_self">Th2</a>, <a class="ref" href="zfmisc_1.html#T129">ZFMISC_1:129</a></i>;<br/><b>hence </b><a NAME="E12:54_1"/>
contradiction
 ;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E4:54"/>
<font color="Maroon">t</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>
 ;<br/>

<b>then reconsider </b><font color="Maroon">t</font> = <font color="Maroon">t</font> as    <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> ;<br/>
<a NAME="E6:54"/>
<font color="Maroon">t</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">r</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:54"/><i><font color="Green">E53</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">r</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">t</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">r</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><a class="ref" href="instalg1.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E10:54"/>
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S1</font>
 
<b>by </b><i><a class="ref" href="instalg1.html#T2" target="_self">Th2</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">a</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="E12:54"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">t</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">a</font>
 <b>by </b><i><a class="ref" href="instalg1.html#T2" target="_self">Th2</a>, <a class="ref" href="msaterm.html#T10">MSATERM:10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</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">b</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/>
<a NAME="E15:54"/>
<font color="Maroon">t</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="trees_4.html#D4">TREES_4:def 4</a></i>;<br/>
<a NAME="E16:54"/><b>then </b><i><font color="Green">E57</font></i>: 
 <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">t</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#T17">MSATERM:17</a></i>;<br/>
<a NAME="E17:54"/><i><font color="Green">E81</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="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2"><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">a</font></span>)</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> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">b</font>
 
<b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</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="E19:54"/>
<span class="p1">(<span class="default"><a href="msaterm.html#K2">Sym</a> <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>,<font color="Maroon">X</font></span>)</span> <a href="msaterm.html#K6">-tree</a> <font color="Maroon">b</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> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="msafree.html#D11">MSAFREE:def 11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">T</font> = <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> as   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S2</font>,<font color="Maroon">X</font> ;<br/>
<a NAME="E21:54"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">T</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"><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>
 
<b>by </b><i><a class="ref" href="trees_4.html#D4">TREES_4:def 4</a></i>;<br/>
<a NAME="E22:54"/>
<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="zfmisc_1.html#T129">ZFMISC_1:129</a></i>;<br/>
<b>hence </b><a NAME="E23:54"/>
<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">t</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</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/>


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