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<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">s1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> st <br/>( <font color="Olive">s1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> <a href="msualg_4.html#K2">.</a> <font color="Olive">s1</font> );<br/>
<a NAME="E1:32_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">E</font> is <a href="osalg_4.html#V1" target="_self">os-compatible</a>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_1_1_1"/>
( the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="osalg_4.html#K4" target="_self">-carrier_of</a> <font color="Maroon">C</font> is <a href="xboole_0.html#V1">empty</a> or  not the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="osalg_4.html#K4" target="_self">-carrier_of</a> <font color="Maroon">C</font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_1"><b>suppose </b></a><a NAME="E1:32_1_1_1_1"/><i><font color="Green">E42</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="osalg_4.html#K4" target="_self">-carrier_of</a> <font color="Maroon">C</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:32_1_1_1_1"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="xboole_0.html#K1">{}</a>  iff  ex <font color="Olive">s1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> st <br/>( <font color="Olive">s1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> <a href="msualg_4.html#K2">.</a> <font color="Olive">s1</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_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> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>thus </b><a NAME="E1:32_1_1_1_1_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="xboole_0.html#K1">{}</a>  implies  ex <font color="Olive">s1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> st <br/>( <font color="Olive">s1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> <a href="msualg_4.html#K2">.</a> <font color="Olive">s1</font> ) )
 ;<br/>

<b>assume </b><a NAME="E2:32_1_1_1_1_1"/>
 ex <font color="Olive">s1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> st <br/>( <font color="Olive">s1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> <a href="msualg_4.html#K2">.</a> <font color="Olive">s1</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">s1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E4:32_1_1_1_1_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">s1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">s1</font> )
 ;<br/>
<a NAME="E5:32_1_1_1_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E6:32_1_1_1_1_1"/>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font></span>)</span> where <font color="Olive">s</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> : <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E7:32_1_1_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font></span>)</span> where <font color="Olive">s</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> : <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> </span>}</span> 
 
<b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>hence </b><a NAME="E8:32_1_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:32_1_1_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being  <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="osalg_4.html#K4" target="_self">-carrier_of</a> <font color="Maroon">C</font> st <br/>for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> iff  ex <font color="Olive">s1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> st <br/>( <font color="Olive">s1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> <a href="msualg_4.html#K2">.</a> <font color="Olive">s1</font> ) )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_2"><b>suppose </b></a><a NAME="E1:32_1_1_1_2"/>
not the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="osalg_4.html#K4" target="_self">-carrier_of</a> <font color="Maroon">C</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">SAC</font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="osalg_4.html#K4" target="_self">-carrier_of</a> <font color="Maroon">C</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">CC</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">SAC</font>, <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">SAC</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] </span>}</span> ;<br/><a NAME="E3:32_1_1_1_2"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">SAC</font>, <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">SAC</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] </span>}</span>  iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_1_2_1"><b>proof </b></a><div class="add">

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

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:32_1_1_1_2_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">SAC</font>, <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">SAC</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">x1</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">SAC</font>, <font color="Maroon">y1</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">SAC</font><b> such that </b><br/><a NAME="E3:32_1_1_1_2_1_1"/><i><font color="Green">E62</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">y1</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E4:32_1_1_1_2_1_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x1</font>,<font color="Maroon">y1</font>]
 ;<br/><b>thus </b><a NAME="E5:32_1_1_1_2_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y</font>]
 <b>by </b><i>, </i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:32_1_1_1_2_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y</font>]
 ;<br/>

<b>then consider </b><font color="Maroon">s1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E4:32_1_1_1_2_1"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">s1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">s1</font> )
 ;<br/>
<a NAME="E5:32_1_1_1_2_1"/><i><font color="Green">E70</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font></span>)</span> where <font color="Olive">s</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> : <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E6:32_1_1_1_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> )
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E7:32_1_1_1_2_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font></span>)</span> where <font color="Olive">s</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> : <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> </span>}</span>  &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font></span>)</span> where <font color="Olive">s</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> : <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> </span>}</span>  )
 
<b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font>, <font color="Maroon">y1</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">SAC</font> ;<br/>
<a NAME="E9:32_1_1_1_2_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">y1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">SAC</font>, <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">SAC</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E10:32_1_1_1_2_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">SAC</font>, <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">SAC</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] </span>}</span> 
 ;<br/>


</div><b>end;</b></div><a NAME="E4:32_1_1_1_2"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">SAC</font>, <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">SAC</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">SAC</font>,<font color="Maroon">SAC</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</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="E1:32_1_1_1_2_2"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">SAC</font>, <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">SAC</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] </span>}</span> 
 ;<br/>

<b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">SAC</font>, <font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">SAC</font><b> such that </b><br/><a NAME="E3:32_1_1_1_2_2"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y</font>] )
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E4:32_1_1_1_2_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">SAC</font>,<font color="Maroon">SAC</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">P1</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">SAC</font>, <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">SAC</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] </span>}</span>  as   <a href="relset_1.html#NM3">Relation</a> of <font color="Maroon">SAC</font> <b>by </b><i><a class="ref" href="relset_1.html#D1">RELSET_1:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_1_2_3"><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="E1:32_1_1_1_2_3"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SAC</font>
 ;<br/><b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">SAC</font> <b>by </b><i/>;<br/><a NAME="E3:32_1_1_1_2_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font></span>)</span> where <font color="Olive">s</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> : <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> </span>}</span> 
 <b>by </b><i/>;<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="E5:32_1_1_1_2_3"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E6:32_1_1_1_2_3"/><i><font color="Green">E86</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">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font></span>)</span> where <font color="Olive">s</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> : <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">s</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E8:32_1_1_1_2_3"/><i><font color="Green">E87</font></i>: 
( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> &amp; <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 <b>by </b><i/>;<br/><a NAME="E9:32_1_1_1_2_3"/>
 <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="msualg_4.html#K2">.</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">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="orders_1.html#T97">ORDERS_1:97</a></i>;<br/><a NAME="E10:32_1_1_1_2_3"/><b>then </b>
<font color="Maroon">E</font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">s</font> <a href="relat_2.html#R1">is_reflexive_in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="relat_2.html#D9">RELAT_2:def 9</a></i>;<br/><a NAME="E11:32_1_1_1_2_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">s</font>
 <b>by </b><i>, , <a class="ref" href="relat_2.html#D1">RELAT_2:def 1</a></i>;<br/><a NAME="E12:32_1_1_1_2_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">SAC</font>, <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">SAC</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] </span>}</span> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E13:32_1_1_1_2_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font>
 ;<br/></div><b>end;</b></div><a NAME="E7:32_1_1_1_2"/><b>then </b>
<font color="Maroon">P1</font> <a href="relat_2.html#R1">is_reflexive_in</a> <font color="Maroon">SAC</font>
 <b>by </b><i><a class="ref" href="relat_2.html#D1">RELAT_2:def 1</a></i>;<br/><a NAME="E8:32_1_1_1_2"/><b>then </b><i><font color="Green">E88</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">SAC</font> &amp;  <a href="relat_1.html#K3">field</a> <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">SAC</font> )
 <b>by </b><i><a class="ref" href="orders_1.html#T98">ORDERS_1:98</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_1_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:32_1_1_1_2_4"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SAC</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SAC</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font> )
 ;<br/><b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font>, <font color="Maroon">y1</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">SAC</font> <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">x2</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">SAC</font>, <font color="Maroon">y2</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">SAC</font><b> such that </b><br/><a NAME="E4:32_1_1_1_2_4"/><i><font color="Green">E91</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x2</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x2</font>,<font color="Maroon">y2</font>] )
 <b>by </b><i/>;<br/><a NAME="E5:32_1_1_1_2_4"/><i><font color="Green">E92</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font> )
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>consider </b><font color="Maroon">s5</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E7:32_1_1_1_2_4"/><i><font color="Green">E94</font></i>: 
( <font color="Maroon">s5</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x2</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">s5</font> )
 <b>by </b><i/>;<br/><a NAME="E8:32_1_1_1_2_4"/><i><font color="Green">E95</font></i>: 
( <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s5</font> &amp; <font color="Maroon">y2</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s5</font> )
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E9:32_1_1_1_2_4"/>
 <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">s5</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s5</font>
 <b>by </b><i><a class="ref" href="orders_1.html#T97">ORDERS_1:97</a></i>;<br/><a NAME="E10:32_1_1_1_2_4"/><b>then </b>
<font color="Maroon">E</font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">s5</font> <a href="relat_2.html#R3">is_symmetric_in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s5</font>
 <b>by </b><i><a class="ref" href="relat_2.html#D11">RELAT_2:def 11</a></i>;<br/><a NAME="E11:32_1_1_1_2_4"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">s5</font>
 <b>by </b><i>, , , <a class="ref" href="relat_2.html#D3">RELAT_2:def 3</a></i>;<br/><a NAME="E12:32_1_1_1_2_4"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">x1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">SAC</font>, <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">SAC</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] </span>}</span> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E13:32_1_1_1_2_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font>
 ;<br/></div><b>end;</b></div><a NAME="E10:32_1_1_1_2"/><b>then </b><i><font color="Green">E96</font></i>: 
<font color="Maroon">P1</font> <a href="relat_2.html#R3">is_symmetric_in</a> <font color="Maroon">SAC</font>
 <b>by </b><i><a class="ref" href="relat_2.html#D3">RELAT_2:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_1_2_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:32_1_1_1_2_5"/><i><font color="Green">E97</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SAC</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SAC</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SAC</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font> )
 ;<br/><b>consider </b><font color="Maroon">x2</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">SAC</font>, <font color="Maroon">y2</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">SAC</font><b> such that </b><br/><a NAME="E3:32_1_1_1_2_5"/><i><font color="Green">E100</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x2</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x2</font>,<font color="Maroon">y2</font>] )
 <b>by </b><i/>;<br/><a NAME="E4:32_1_1_1_2_5"/><i><font color="Green">E101</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font> )
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>consider </b><font color="Maroon">y3</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">SAC</font>, <font color="Maroon">z3</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">SAC</font><b> such that </b><br/><a NAME="E6:32_1_1_1_2_5"/><i><font color="Green">E102</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y3</font>,<font color="Maroon">z3</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">y3</font>,<font color="Maroon">z3</font>] )
 <b>by </b><i/>;<br/><a NAME="E7:32_1_1_1_2_5"/><i><font color="Green">E103</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y3</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z3</font> )
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>consider </b><font color="Maroon">s5</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E9:32_1_1_1_2_5"/><i><font color="Green">E104</font></i>: 
( <font color="Maroon">s5</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x2</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">s5</font> )
 <b>by </b><i/>;<br/><a NAME="E10:32_1_1_1_2_5"/><i><font color="Green">E105</font></i>: 
( <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s5</font> &amp; <font color="Maroon">y2</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s5</font> )
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>consider </b><font color="Maroon">s6</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E12:32_1_1_1_2_5"/><i><font color="Green">E106</font></i>: 
( <font color="Maroon">s6</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y3</font>,<font color="Maroon">z3</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">s6</font> )
 <b>by </b><i/>;<br/><a NAME="E13:32_1_1_1_2_5"/><i><font color="Green">E107</font></i>: 
( <font color="Maroon">y3</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s6</font> &amp; <font color="Maroon">z3</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s6</font> )
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>reconsider </b><font color="Maroon">s51</font> = <font color="Maroon">s5</font>, <font color="Maroon">s61</font> = <font color="Maroon">s6</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> ;<br/><b>consider </b><font color="Maroon">su</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E16:32_1_1_1_2_5"/><i><font color="Green">E108</font></i>: 
( <font color="Maroon">su</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Maroon">s51</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">su</font> &amp; <font color="Maroon">s61</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">su</font> )
 <b>by </b><i>, , <a class="ref" href="waybel_0.html#D1">WAYBEL_0:def 1</a></i>;<br/><a NAME="E17:32_1_1_1_2_5"/><i><font color="Green">E109</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x2</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">su</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y3</font>,<font color="Maroon">z3</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">su</font> )
 <b>by </b><i>, , , , , , </i>;<br/><a NAME="E18:32_1_1_1_2_5"/><b>then </b><i><font color="Green">E110</font></i>: 
( <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">su</font> &amp; <font color="Maroon">y2</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">su</font> &amp; <font color="Maroon">z3</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">su</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E19:32_1_1_1_2_5"/>
 <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">su</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">su</font>
 <b>by </b><i><a class="ref" href="orders_1.html#T97">ORDERS_1:97</a></i>;<br/><a NAME="E20:32_1_1_1_2_5"/><b>then </b>
<font color="Maroon">E</font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">su</font> <a href="relat_2.html#R8">is_transitive_in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">su</font>
 <b>by </b><i><a class="ref" href="relat_2.html#D16">RELAT_2:def 16</a></i>;<br/><a NAME="E21:32_1_1_1_2_5"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x2</font>,<font color="Maroon">z3</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">su</font>
 <b>by </b><i>, , , , <a class="ref" href="relat_2.html#D8">RELAT_2:def 8</a></i>;<br/><b>hence </b><a NAME="E22:32_1_1_1_2_5"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><a NAME="E12:32_1_1_1_2"/><b>then </b>
<font color="Maroon">P1</font> <a href="relat_2.html#R8">is_transitive_in</a> <font color="Maroon">SAC</font>
 <b>by </b><i><a class="ref" href="relat_2.html#D8">RELAT_2:def 8</a></i>;<br/><b>then reconsider </b><font color="Maroon">P1</font> = <font color="Maroon">P1</font> as   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="osalg_4.html#K4" target="_self">-carrier_of</a> <font color="Maroon">C</font> <b>by </b><i>, , <a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a>, <a class="ref" href="relat_2.html#D11">RELAT_2:def 11</a>, <a class="ref" href="relat_2.html#D16">RELAT_2:def 16</a></i>;<br/><b>take </b>
<font color="Maroon">P1</font>
;<br/><b>thus </b><a NAME="E14:32_1_1_1_2"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font> iff  ex <font color="Olive">s1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> st <br/>( <font color="Olive">s1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> <a href="msualg_4.html#K2">.</a> <font color="Olive">s1</font> ) )
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
