<?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">U0</font> be    <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">B</font> be   <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">U0</font>;<br/>





<b>assume </b><a NAME="E1:17"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">B</font> <a href="pboole.html#R6">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U0</font>
 ;<br/>

<b>thus </b><a NAME="E2:17"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">B</font> is <a href="msualg_2.html#V3" target="_self">opers_closed</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><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" target="_self">MSUALG_2:def 7</a><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">B</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> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <span class="p1">(<span class="default"><font color="Maroon">B</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> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1"><b>suppose </b></a><a NAME="E1:17_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">B</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> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:17_1_1_1"/><b>then </b><i><font color="Green">E22</font></i>: 
 <a href="msualg_1.html#K4">Result</a> <font color="Maroon">o</font>,<font color="Maroon">U0</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="msualg_1.html#D10">MSUALG_1:def 10</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1_1_1_1_1"/>
(  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<font color="Maroon">U0</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<font color="Maroon">U0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:17_1_1_1_1_1_1"/>
 <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<font color="Maroon">U0</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:17_1_1_1_1_1_1"/>
 <a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">U0</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="partfun1.html#T57">PARTFUN1:57</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_1_1_1_1_2"/>
 <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<font color="Maroon">U0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:17_1_1_1_1_1_2"/>
 <a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">U0</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E4:17_1_1_1"/><b>then </b>
 <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">U0</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">B</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="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="relat_1.html#T60">RELAT_1:60</a>, <a class="ref" href="relat_1.html#T111">RELAT_1:111</a></i>;<br/><b>hence </b><a NAME="E5:17_1_1_1"/>
 <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">U0</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">B</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="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">B</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="xboole_1.html#T2">XBOOLE_1:2</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_2.html#D6" target="_self">MSUALG_2:def 6</a><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">B</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> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:17_1_1_2"/><b>then </b>
 <a href="msualg_1.html#K4">Result</a> <font color="Maroon">o</font>,<font color="Maroon">U0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="msualg_1.html#D10">MSUALG_1:def 10</a></i>;<br/><a NAME="E3:17_1_1_2"/><b>then </b><i><font color="Green">E26</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">U0</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<font color="Maroon">U0</font> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">U0</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="msualg_1.html#K4">Result</a> <font color="Maroon">o</font>,<font color="Maroon">U0</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#T12">RELSET_1:12</a></i>;<br/><a NAME="E4:17_1_1_2"/><i><font color="Green">E27</font></i>: 
 <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<font color="Maroon">U0</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">B</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="msualg_1.html#D9">MSUALG_1:def 9</a></i>;<br/><a NAME="E5:17_1_1_2"/>
 <a href="msualg_1.html#K4">Result</a> <font color="Maroon">o</font>,<font color="Maroon">U0</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">B</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/><b>hence </b><a NAME="E6:17_1_1_2"/>
 <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">U0</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">B</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="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">B</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="relat_1.html#T97">RELAT_1:97</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_2.html#D6" target="_self">MSUALG_2:def 6</a><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>

<b>let </b><font color="Maroon">o</font> be   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font>;<br/>

<a NAME="E3:17"/>
<font color="Maroon">B</font> <a href="msualg_2.html#R1" target="_self">is_closed_on</a> <font color="Maroon">o</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:17"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">o</font> <a href="msualg_2.html#K3" target="_self">/.</a> <font color="Maroon">B</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">U0</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">B</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>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">B</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> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <span class="p1">(<span class="default"><font color="Maroon">B</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> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_2_1"><b>suppose </b></a><a NAME="E1:17_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">B</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> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:17_2_1"/><b>then </b><i><font color="Green">E29</font></i>: 
 <a href="msualg_1.html#K4">Result</a> <font color="Maroon">o</font>,<font color="Maroon">U0</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="msualg_1.html#D10">MSUALG_1:def 10</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_2_1_1_1"/>
(  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<font color="Maroon">U0</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<font color="Maroon">U0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:17_2_1_1_1_1"/>
 <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<font color="Maroon">U0</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:17_2_1_1_1_1"/>
 <a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">U0</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="partfun1.html#T57">PARTFUN1:57</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:17_2_1_1_1_2"/>
 <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<font color="Maroon">U0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:17_2_1_1_1_2"/>
 <a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">U0</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:17_2_1"/>
<font color="Maroon">o</font> <a href="msualg_2.html#K3" target="_self">/.</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">U0</font>
 <b>by </b><i>, <a class="ref" href="relat_1.html#T111">RELAT_1:111</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_2_2"><b>suppose </b></a><a NAME="E1:17_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon">B</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> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:17_2_2"/><b>then </b>
 <a href="msualg_1.html#K4">Result</a> <font color="Maroon">o</font>,<font color="Maroon">U0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="msualg_1.html#D10">MSUALG_1:def 10</a></i>;<br/><a NAME="E3:17_2_2"/><b>then </b><i><font color="Green">E30</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">U0</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<font color="Maroon">U0</font> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">U0</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="msualg_1.html#K4">Result</a> <font color="Maroon">o</font>,<font color="Maroon">U0</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#T12">RELSET_1:12</a></i>;<br/><a NAME="E4:17_2_2"/>
 <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<font color="Maroon">U0</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">B</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="msualg_1.html#D9">MSUALG_1:def 9</a></i>;<br/><b>hence </b><a NAME="E5:17_2_2"/>
<font color="Maroon">o</font> <a href="msualg_2.html#K3" target="_self">/.</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">U0</font>
 <b>by </b><i>, , <a class="ref" href="relat_1.html#T97">RELAT_1:97</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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