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

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:8_1_1_1"/>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>then reconsider </b><font color="Maroon">A</font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/><b>set </b><font color="Maroon">E</font> = <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> :  ex <font color="Olive">o</font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font> st <br/>( the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">o</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp; <font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Olive">o</font>,<font color="Maroon">U0</font></span>)</span> ) </span>}</span> ;<br/><a NAME="E3:8_1_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> :  ex <font color="Olive">o</font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font> st <br/>( the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">o</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp; <font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Olive">o</font>,<font color="Maroon">U0</font></span>)</span> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1_1"><b>proof </b></a><div class="add">

<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="E1:8_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> :  ex <font color="Olive">o</font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font> st <br/>( the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">o</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp; <font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Olive">o</font>,<font color="Maroon">U0</font></span>)</span> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">w</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font><b> such that </b><br/><a NAME="E3:8_1_1_1_1"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp;  ex <font color="Olive">o</font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font> st <br/>( the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">o</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Olive">o</font>,<font color="Maroon">U0</font></span>)</span> ) )
 ;<br/>
<b>thus </b><a NAME="E4:8_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">E</font> = <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> :  ex <font color="Olive">o</font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font> st <br/>( the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">o</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp; <font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Olive">o</font>,<font color="Maroon">U0</font></span>)</span> ) </span>}</span>  as   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font></span>)</span> ;<br/><b>take </b><font color="Maroon">E</font> = <font color="Maroon">E</font>;<br/><b>take </b><font color="Maroon">A</font> = <font color="Maroon">A</font>;<br/><b>thus </b><a NAME="E5:8_1_1_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> &amp; <font color="Maroon">E</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> :  ex <font color="Olive">o</font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font> st <br/>( the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">o</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp; <font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Olive">o</font>,<font color="Maroon">U0</font></span>)</span> ) </span>}</span>  )
 ;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:8_1_1"/>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>take </b>
 <a href="subset_1.html#K1">{}</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font></span>)</span>
;<br/>

<b>thus </b><a NAME="E3:8_1_1"/>
 <a href="subset_1.html#K1">{}</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>


</div>
