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

<b>let </b><font color="Maroon">S</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">U1</font> be  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">U2</font> be  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">Gen</font> be    <a href="msafree.html#M1">GeneratorSet</a> of <font color="Maroon">U1</font>;<br/><b>let </b><font color="Maroon">h1</font> be   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">U1</font>,<font color="Maroon">U2</font>, <font color="Maroon">h2</font> be   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">U1</font>,<font color="Maroon">U2</font>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:22"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">h1</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">U1</font>,<font color="Maroon">U2</font>
 <b>and </b><br/><a NAME="E2:22"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">h2</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">U1</font>,<font color="Maroon">U2</font>
 <b>and </b><br/><a NAME="E3:22"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">h1</font> <a href="msafree.html#K1">||</a> <font color="Maroon">Gen</font> <a href="pboole.html#R6">=</a> <font color="Maroon">h2</font> <a href="msafree.html#K1">||</a> <font color="Maroon">Gen</font>
 ;<br/>

<b>set </b><font color="Maroon">I</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</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> ] <b>means </b> ex <font color="Olive">s</font> being  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">s</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="msualg_3.html#K1">.</a> <font color="Olive">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="msualg_3.html#K1">.</a> <font color="Olive">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>2</sub></font> );<br/>
<b>consider </b><font color="Maroon">A</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="E5:22"/><i><font color="Green">E53</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/>for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <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="Olive">i</font> iff ( <font color="Olive">e</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>,<font color="Olive">e</font>] ) )
 <b>from </b><i><a class="ref" href="pboole.html#S2">PBOOLE:sch 2</a>();<br/></i>
<a NAME="E6:22"/>
<font color="Maroon">A</font> is    <a href="pboole.html#M4">ManySortedSubset</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>proof </b></a><div class="add">

<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="pboole.html#D5">PBOOLE:def 5</a>,<a class="ref" href="pboole.html#D23">PBOOLE:def 23</a><br/>

<b>assume </b><a NAME="E1:22_1"/><i><font color="Green">E54</font></i>: 
<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>let </b><font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:22_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">i</font>
 ;<br/>

<b>hence </b><a NAME="E3:22_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">A</font> = <font color="Maroon">A</font> as   <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">U1</font> ;<br/>
<a NAME="E8:22"/>
<font color="Maroon">A</font> is <a href="msualg_2.html#V3">opers_closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_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">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_2.html#D7">MSUALG_2:def 7</a><br/><b>let </b><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="msualg_2.html#D6">MSUALG_2:def 6</a>,<a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>



<b>set </b><font color="Maroon">r</font> =  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font>;<br/>
<b>set </b><font color="Maroon">ar</font> =  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font>;<br/>
<b>assume </b><a NAME="E1:22_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">U1</font></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">A</font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font></span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:22_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">U1</font></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">A</font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E4:22_2"/><i><font color="Green">E60</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">U1</font></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">A</font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:22_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T86">RELAT_1:86</a></i>;<br/>
<a NAME="E6:22_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>;<br/>
<a NAME="E7:22_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="funct_1.html#K1">.</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="msualg_1.html#D6">MSUALG_1:def 6</a></i>;<br/>
<a NAME="E8:22_2"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pboole.html#D19">PBOOLE:def 19</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<font color="Maroon">U1</font> <b>by </b><i/>;<br/>
<a NAME="E10:22_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">U1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<a NAME="E11:22_2"/>
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">U1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 
<b>by </b><i><a class="ref" href="msualg_1.html#D10">MSUALG_1:def 10</a></i>;<br/>
<a NAME="E12:22_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">U1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> is    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>;<br/>
<a NAME="E13:22_2"/><b>then </b><i><font color="Green">E67</font></i>: 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">U1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> is    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span>
 
<b>by </b><i><a class="ref" href="msualg_1.html#D7">MSUALG_1:def 7</a></i>;<br/>
<a NAME="E14:22_2"/><i><font color="Green">E73</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">x</font></span>)</span> <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="msualg_3.html#T6">MSUALG_3:6</a></i>;<br/>
<a NAME="E15:22_2"/><i><font color="Green">E74</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">x</font></span>)</span> <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="msualg_3.html#T6">MSUALG_3:6</a></i>;<br/>
<a NAME="E16:22_2"/><i><font color="Green">E75</font></i>: 
for <font color="Olive">n</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">x</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:22_2_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">x</font></span>)</span>
 ;<br/>

<a NAME="E2:22_2_1"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">n'</font> = <font color="Maroon">n</font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E4:22_2_1"/><i><font color="Green">E77</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">x</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="msualg_3.html#T6">MSUALG_3:6</a></i>;<br/>
<a NAME="E5:22_2_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<a NAME="E6:22_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 
<b>by </b><i>, , , , <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<a NAME="E7:22_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>
<a NAME="E8:22_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<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">n</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E9:22_2_1"/><b>then </b>
 ex <font color="Olive">s</font> being  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font> st <br/>( <font color="Olive">s</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">n</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="msualg_3.html#K1">.</a> <font color="Olive">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="msualg_3.html#K1">.</a> <font color="Olive">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> )
 
<b>by </b><i/>;<br/>
<b>hence </b><span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<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">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n'</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="msualg_3.html#D8">MSUALG_3:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
<b>by </b><i>, , , <a class="ref" href="msualg_3.html#D8">MSUALG_3:def 8</a></i>
;<br/><br/>


</div><b>end;</b></div>
<span class="p1">(<span class="default"><font color="Maroon">h1</font> <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> <font color="Maroon">y</font> = 
<span class="p1">(<span class="default"><font color="Maroon">h1</font> <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">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">U1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">U2</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, <a class="ref" href="msualg_3.html#D9">MSUALG_3:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">U2</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">h2</font> <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">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">U1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, <a class="ref" href="msualg_3.html#D9">MSUALG_3:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">h2</font> <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> <font color="Maroon">y</font>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/>
<a NAME="E18:22_2"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E19:22_2"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font></span>)</span>
 
<b>by </b><i><a class="ref" href="msualg_1.html#D7">MSUALG_1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E20:22_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:22"/><b>then </b><i><font color="Green">E78</font></i>: 
<font color="Maroon">U1</font> <a href="msualg_2.html#K10">|</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#G3">MSAlgebra</a>(# <font color="Maroon">A</font>,<span class="p1">(<span class="default"><a href="msualg_2.html#K4">Opers</a> <font color="Maroon">U1</font>,<font color="Maroon">A</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="msualg_2.html#D16">MSUALG_2:def 16</a></i>;<br/>
<a NAME="E10:22"/>
<font color="Maroon">Gen</font> is    <a href="pboole.html#M4">ManySortedSubset</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon">U1</font> <a href="msualg_2.html#K10">|</a> <font color="Maroon">A</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_3"><b>proof </b></a><div class="add">

<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="pboole.html#D5">PBOOLE:def 5</a>,<a class="ref" href="pboole.html#D23">PBOOLE:def 23</a><br/>

<b>assume </b><a NAME="E1:22_3"/><i><font color="Green">E79</font></i>: 
<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>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:22_3"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Gen</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/>

<a NAME="E3:22_3"/>
<font color="Maroon">Gen</font> <a href="pboole.html#R2">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E4:22_3"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">Gen</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i>, <a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/>
<b>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> <b>by </b><i/>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">h1</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">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</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">Gen</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">x</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"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="msafree.html#K1">||</a> <font color="Maroon">Gen</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">x</font>
<b>by </b><i><a class="ref" href="msafree.html#D1">MSAFREE:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h2</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">Gen</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">x</font>
<b>by </b><i>, <a class="ref" href="msafree.html#D1">MSAFREE:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">h2</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">x</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/>
<b>hence </b><a NAME="E7:22_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon">U1</font> <a href="msualg_2.html#K10">|</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:22"/><b>then </b><i><font color="Green">E82</font></i>: 
 <a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">Gen</font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">U1</font> <a href="msualg_2.html#K10">|</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="msualg_2.html#D18">MSUALG_2:def 18</a></i>;<br/>
<a NAME="E12:22"/>
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">Gen</font></span>)</span> <a href="pboole.html#R6">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font>
 
<b>by </b><i><a class="ref" href="msafree.html#D4">MSAFREE:def 4</a></i>;<br/>
<a NAME="E13:22"/><b>then </b>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> is    <a href="pboole.html#M4">ManySortedSubset</a> of <font color="Maroon">A</font>
 
<b>by </b><i>, , <a class="ref" href="msualg_2.html#D10">MSUALG_2:def 10</a></i>;<br/>
<a NAME="E14:22"/><b>then </b><i><font color="Green">E83</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_4"><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="E1:22_4"/><i><font color="Green">E84</font></i>: 
<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/><a NAME="E3:22_4"/><i><font color="Green">E85</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E4:22_4"/><i><font color="Green">E86</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_4_1"><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="E1:22_4_1"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span>
 ;<br/><a NAME="E2:22_4_1"/>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , <a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/><a NAME="E3:22_4_1"/><b>then </b>
 ex <font color="Olive">t</font> being  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font> st <br/>( <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="msualg_3.html#K1">.</a> <font color="Olive">t</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="msualg_3.html#K1">.</a> <font color="Olive">t</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E4:22_4_1"/>
<span class="p1">(<span class="default"><font color="Maroon">h1</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">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h2</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">x</font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:22_4"/>
<font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E16:22"/>
<font color="Maroon">h1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">h2</font>
 <b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


</div>
