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

<b>set </b><font color="Maroon">c</font> =  <a href="osalg_1.html#K5" target="_self">ConstOSSet</a> <font color="Maroon">S</font>,<font color="Maroon">z</font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font>) <b>-&gt; </b>   <a href="relset_1.html#M2">Relation</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osalg_1.html#K5" target="_self">ConstOSSet</a> <font color="Maroon">S</font>,<font color="Maroon">z</font></span>)</span> <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">a<sub>1</sub></font>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">z1</font></span><a href="tarski.html#K1">}</a></span> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="osalg_1.html#K5" target="_self">ConstOSSet</a> <font color="Maroon">S</font>,<font color="Maroon">z</font></span>)</span> <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">a<sub>1</sub></font></span>)</span> <a href="pboole.html#K9">--&gt;</a> <font color="Maroon">z1</font>;<br/>
<b>consider </b><font color="Maroon">ch2</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E2:78_1_1"/><i><font color="Green">E24</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">ch2</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> &amp; ( for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> holds  <font color="Maroon">ch2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>) ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S4">FUNCT_1:sch 4</a>();<br/></i>
<b>reconsider </b><font color="Maroon">ch4</font> = <font color="Maroon">ch2</font> as    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="osalg_1.html#D4" target="_self">Def4</a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E4:78_1_1"/>
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="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> holds <br/><font color="Maroon">ch4</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osalg_1.html#K5" target="_self">ConstOSSet</a> <font color="Maroon">S</font>,<font color="Maroon">z</font></span>)</span> <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="Olive">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_1.html#K5" target="_self">ConstOSSet</a> <font color="Maroon">S</font>,<font color="Maroon">z</font></span>)</span> <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="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_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="E1:78_1_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">o</font> = <font color="Maroon">i</font> as   <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="osalg_1.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E3:78_1_1_1"/>
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> <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="E4:78_1_1_1"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="relset_1.html#K11">"</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_1.html#K5" target="_self">ConstOSSet</a> <font color="Maroon">S</font>,<font color="Maroon">z</font></span>)</span> <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> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="relset_1.html#K11">"</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> <a href="pboole.html#K9">--&gt;</a> <font color="Maroon">z</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T25">FUNCOP_1:25</a></i>
<br/>.= 
<font color="Maroon">z</font>
<b>by </b><i><a class="ref" href="osalg_1.html#T3" target="_self">Th3</a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/>
<a NAME="E6:78_1_1_1"/><b>then </b><i><font color="Green">E53</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">z1</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_1.html#K5" target="_self">ConstOSSet</a> <font color="Maroon">S</font>,<font color="Maroon">z</font></span>)</span> <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">i</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E7:78_1_1_1"/>
<font color="Maroon">ch4</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">o</font>)
 
<b>by </b><i><a class="ref" href="osalg_1.html#D4" target="_self">Def4</a></i>;<br/>
<b>hence </b><a NAME="E8:78_1_1_1"/>
<font color="Maroon">ch4</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osalg_1.html#K5" target="_self">ConstOSSet</a> <font color="Maroon">S</font>,<font color="Maroon">z</font></span>)</span> <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">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_1.html#K5" target="_self">ConstOSSet</a> <font color="Maroon">S</font>,<font color="Maroon">z</font></span>)</span> <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">i</font>
 <b>by </b><i>, <a class="ref" href="funct_2.html#T9">FUNCT_2:9</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">ch3</font> = <font color="Maroon">ch4</font> as    <a href="pboole.html#M3">ManySortedFunction</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_1.html#K5" target="_self">ConstOSSet</a> <font color="Maroon">S</font>,<font color="Maroon">z</font></span>)</span> <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 class="p1">(<span class="default"><a href="osalg_1.html#K5" target="_self">ConstOSSet</a> <font color="Maroon">S</font>,<font color="Maroon">z</font></span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/>
<b>take </b><font color="Maroon">T</font> =  <a href="osalg_1.html#K6" target="_self">ConstOSA</a> <font color="Maroon">S</font>,<font color="Maroon">z</font>,<font color="Maroon">ch3</font>;<br/>

<b>thus </b><a NAME="E6:78_1_1"/><i><font color="Green">E56</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">T</font> <a href="pboole.html#R6">=</a>  <a href="osalg_1.html#K5" target="_self">ConstOSSet</a> <font color="Maroon">S</font>,<font color="Maroon">z</font>
 <b>by </b><i/>;<br/>

<b>let </b><font color="Maroon">o</font> be   <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font>;<br/>

 <a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">T</font> = 
the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
<b>by </b><i><a class="ref" href="msualg_1.html#D11">MSUALG_1:def 11</a></i>
<br/>.= 
<font color="Maroon">ch3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="osalg_1.html#K5" target="_self">ConstOSSet</a> <font color="Maroon">S</font>,<font color="Maroon">z</font></span>)</span> <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> <a href="pboole.html#K9">--&gt;</a> <font color="Maroon">z1</font>
<b>by </b><i><a class="ref" href="osalg_1.html#D4" target="_self">Def4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<font color="Maroon">T</font></span>)</span> <a href="pboole.html#K9">--&gt;</a> <font color="Maroon">z1</font>
<b>by </b><i><a class="ref" href="osalg_1.html#D5" target="_self">Def5</a>, <a class="ref" href="msualg_1.html#D9">MSUALG_1:def 9</a></i>
;<br/>
<b>hence </b><a NAME="E8:78_1_1"/>
 <a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">T</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<font color="Maroon">T</font></span>)</span> <a href="pboole.html#K9">--&gt;</a> <font color="Maroon">z1</font>
 ;<br/>


</div>
