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

<b>let </b><font color="Maroon">T</font> be    <a href="struct_0.html#L1">1-sorted</a> ;<br/><b>let </b><font color="Maroon">F</font> be  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">F1</font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:76"/><i><font color="Green">E33</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E2:76"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">F1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D8">PRE_TOPC:def 8</a><br/>

<b>set </b><font color="Maroon">ZAW</font> = <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span> ;<br/>
<b>thus </b><a NAME="E3:76"/>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F1</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D8">PRE_TOPC:def 8</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">t</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="E1:76_1"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font>
 ;<br/>

<b>then consider </b><font color="Maroon">Z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:76_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>and </b><br/><a NAME="E4:76_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>set </b><font color="Maroon">ALL</font> = <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Olive">X</font> &amp; <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">X</font> ) </span>}</span> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:76_1_1"/>
( <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Olive">X</font> &amp; <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">X</font> ) </span>}</span>  is <a href="xboole_0.html#V1">empty</a> or  not <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Olive">X</font> &amp; <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">X</font> ) </span>}</span>  is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_1_1"><b>suppose </b></a><a NAME="E1:76_1_1_1"/><i><font color="Green">E46</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Olive">X</font> &amp; <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">X</font> ) </span>}</span>  is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:76_1_1_1_1"/>
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">X</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:76_1_1_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E4:76_1_1_1_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E5:76_1_1_1_1"/><i><font color="Green">E48</font></i>: 
 ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> )
 ;<br/><b>consider </b><font color="Maroon">Y</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E7:76_1_1_1_1"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Y</font> )
 <b>by </b><i><a class="ref" href="rcomp_3.html#T14" target="_self">Th14</a></i>;<br/><a NAME="E8:76_1_1_1_1"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Olive">X</font> &amp; <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">X</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rcomp_3.html#T12" target="_self">Th12</a>, <a class="ref" href="rcomp_3.html#T16" target="_self">Th16</a></i>;<br/><b>hence </b><a NAME="E9:76_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="rcomp_3.html#T7" target="_self">Th7</a></i>;<br/></div><b>end;</b></div><a NAME="E3:76_1_1_1"/><b>then </b>
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F1</font>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T4" target="_self">Th4</a>, <a class="ref" href="rcomp_3.html#T6" target="_self">Th6</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E4:76_1_1_1"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F1</font>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T5" target="_self">Th5</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_1_2"><b>suppose </b></a><a NAME="E1:76_1_1_2"/>
not <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Olive">X</font> &amp; <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">X</font> ) </span>}</span>  is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">w</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:76_1_1_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Olive">X</font> &amp; <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">X</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><b>consider </b><font color="Maroon">R</font> being   <a href="relset_1.html#NM3">Relation</a> of <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Olive">X</font> &amp; <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">X</font> ) </span>}</span> <b> such that </b><br/><a NAME="E5:76_1_1_2"/><i><font color="Green">E51</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> <font color="Maroon">R</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Olive">X</font> &amp; <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">X</font> ) </span>}</span>  &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Olive">X</font> &amp; <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">X</font> ) </span>}</span>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] ) )
 <b>from </b><i><a class="ref" href="relset_1.html#S1">RELSET_1:sch 1</a>();<br/></i><a NAME="E6:76_1_1_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">R</font> <a href="relat_2.html#R1">is_reflexive_in</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Olive">X</font> &amp; <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">X</font> ) </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_2.html#D1">RELAT_2:def 1</a><br/>

<b>thus </b><a NAME="E1:76_1_1_2_1"/>
(  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Olive">X</font> &amp; <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">X</font> ) </span>}</span>  or <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">R</font> )
 <b>by </b><i><a class="ref" href="rcomp_3.html#T12" target="_self">Th12</a></i>;<br/>


</div><b>end;</b></div><a NAME="E7:76_1_1_2"/><i><font color="Green">E53</font></i>: 
 <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Olive">X</font> &amp; <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">X</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rcomp_3.html#T14" target="_self">Th14</a>, <a class="ref" href="orders_1.html#T98">ORDERS_1:98</a></i>;<br/><a NAME="E8:76_1_1_2"/><b>then </b><i><font color="Green">E91</font></i>: 
<font color="Maroon">R</font> is <a href="relat_2.html#V1">reflexive</a>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T14" target="_self">Th14</a>, <a class="ref" href="relat_2.html#D9">RELAT_2:def 9</a></i>;<br/><a NAME="E9:76_1_1_2"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">R</font> <a href="orders_1.html#R2">partially_orders</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Olive">X</font> &amp; <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">X</font> ) </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_1_2_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:76_1_1_2_2"/>
<font color="Maroon">R</font> <a href="relat_2.html#R1">is_reflexive_in</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Olive">X</font> &amp; <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">X</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rcomp_3.html#T14" target="_self">Th14</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="orders_1.html#D7">ORDERS_1:def 7</a><br/>

<b>thus </b><a NAME="E2:76_1_1_2_2"/>
<font color="Maroon">R</font> <a href="relat_2.html#R8">is_transitive_in</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Olive">X</font> &amp; <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">X</font> ) </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_1_2_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> , <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="relat_2.html#D8">RELAT_2:def 8</a><br/>

<b>assume </b><a NAME="E1:76_1_1_2_2_1"/><i><font color="Green">E96</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Olive">X</font> &amp; <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">X</font> ) </span>}</span>  &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Olive">X</font> &amp; <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">X</font> ) </span>}</span>  &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Olive">X</font> &amp; <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">X</font> ) </span>}</span>  )
 ;<br/>

<b>assume </b><a NAME="E2:76_1_1_2_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> <font color="Maroon">R</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">R</font> )
 ;<br/>

<a NAME="E3:76_1_1_2_2_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">z</font> )
 
<b>by </b><i><a class="ref" href="rcomp_3.html#T12" target="_self">Th12</a></i>;<br/>
<a NAME="E4:76_1_1_2_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">z</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E5:76_1_1_2_2_1"/>
<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">R</font>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T19" target="_self">Th19</a>, <a class="ref" href="rcomp_3.html#T12" target="_self">Th12</a></i>;<br/>


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

<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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_2.html#D4">RELAT_2:def 4</a><br/>

<b>assume </b><a NAME="E3:76_1_1_2_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Olive">X</font> &amp; <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">X</font> ) </span>}</span>  &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Olive">X</font> &amp; <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">X</font> ) </span>}</span>  )
 ;<br/>

<b>assume </b><a NAME="E4:76_1_1_2_2"/>
( <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">R</font> &amp; <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">R</font> )
 ;<br/>

<b>hence </b><a NAME="E5:76_1_1_2_2"/>
( <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="rcomp_3.html#T12" target="_self">Th12</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>


</div><b>end;</b></div><a NAME="E10:76_1_1_2"/>
<span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Olive">X</font> &amp; <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">X</font> ) </span>}</span>  <a href="orders_1.html#R4">has_upper_Zorn_property_wrt</a> <font color="Maroon">R</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_1_2_3"><b>proof </b></a><div class="add">

<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="orders_1.html#D9">ORDERS_1:def 9</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:76_1_1_2_3"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Olive">X</font> &amp; <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">X</font> ) </span>}</span> 
 <b>and </b><br/><a NAME="E2:76_1_1_2_3"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">Y</font> <a href="orders_1.html#NR3" target="_self">is_linear-order</a> 
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_1_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:76_1_1_2_3_1"/>
(  not <font color="Maroon">Y</font> is <a href="xboole_0.html#V1">empty</a> or <font color="Maroon">Y</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="76_1_1_2_3_1_1"><b>suppose </b></a><a NAME="E1:76_1_1_2_3_1_1"/><i><font color="Green">E99</font></i>: 
not <font color="Maroon">Y</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:76_1_1_2_3_1_1"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E4:76_1_1_2_3_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Olive">X</font> &amp; <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">X</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rcomp_3.html#T19" target="_self">Th19</a>, <a class="txt" href="rcomp_3.html#E23"><i><font color="Green">Lemma66</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">X</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E6:76_1_1_2_3_1_1"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>and </b><br/><a NAME="E7:76_1_1_2_3_1_1"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E8:76_1_1_2_3_1_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E9:76_1_1_2_3_1_1"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">X</font>
 ;<br/><b>take </b>
 <a href="tarski.html#K3">union</a> <font color="Maroon">Y</font>
;<br/><a NAME="E10:76_1_1_2_3_1_1"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E23"><i><font color="Green">Lemma66</font></i></a>, <a class="ref" href="rcomp_3.html#T21" target="_self">Th21</a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/><a NAME="E11:76_1_1_2_3_1_1"/><b>then </b><i><font color="Green">E105</font></i>: 
<font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T22" target="_self">Th22</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E12:76_1_1_2_3_1_1"/><i><font color="Green">E106</font></i>: 
for <font color="Olive">A</font>, <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> holds <br/><font color="Olive">A</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_1_2_3_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><b>that </b><br/><a NAME="E1:76_1_1_2_3_1_1_1"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E2:76_1_1_2_3_1_1_1"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>

<a NAME="E3:76_1_1_2_3_1_1_1"/>
<font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">Y</font> is <a href="relat_2.html#V6">connected</a>
 
<b>by </b><i><a class="ref" href="rcomp_3.html#T20" target="_self">Th20</a>, <a class="ref" href="orders_1.html#D5">ORDERS_1:def 5</a></i>;<br/>
<a NAME="E4:76_1_1_2_3_1_1_1"/><b>then </b><i><font color="Green">E109</font></i>: 
<font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">Y</font> <a href="relat_2.html#R6">is_connected_in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">Y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_2.html#D14">RELAT_2:def 14</a></i>;<br/>
<a NAME="E5:76_1_1_2_3_1_1_1"/>
 <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">Y</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="rcomp_3.html#T17" target="_self">Th17</a>, <a class="ref" href="rcomp_3.html#T19" target="_self">Th19</a>, <a class="ref" href="rcomp_3.html#T16" target="_self">Th16</a>, <a class="ref" href="orders_1.html#T181">ORDERS_1:181</a></i>;<br/>
<a NAME="E6:76_1_1_2_3_1_1_1"/><b>then </b><i><font color="Green">E110</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">Y</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">A</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">Y</font> or <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> )
 
<b>by </b><i><a class="ref" href="rcomp_3.html#T27" target="_self">Th27</a>, <a class="ref" href="rcomp_3.html#T28" target="_self">Th28</a>, <a class="ref" href="rcomp_3.html#T29" target="_self">Th29</a>, <a class="ref" href="relat_2.html#D6">RELAT_2:def 6</a></i>;<br/>
<a NAME="E7:76_1_1_2_3_1_1_1"/>
<font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E8:76_1_1_2_3_1_1_1"/><b>then </b>
( <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> or <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 
<b>by </b><i><a class="ref" href="rcomp_3.html#T30" target="_self">Th30</a>, <a class="ref" href="rcomp_3.html#T12" target="_self">Th12</a></i>;<br/>
<b>hence </b><a NAME="E9:76_1_1_2_3_1_1_1"/>
<font color="Maroon">A</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T27" target="_self">Th27</a>, <a class="ref" href="rcomp_3.html#T28" target="_self">Th28</a>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<br/>


</div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>( not <font color="Maroon">a<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> implies  <a href="tarski.html#K3">union</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> );<br/><a NAME="E13:76_1_1_2_3_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Olive">X</font> &amp; <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">X</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_1_2_3_1_1_2"><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="E1:76_1_1_2_3_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Olive">X</font> &amp; <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">X</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:76_1_1_2_3_1_1_2"/><b>then </b>
 ex <font color="Olive">X</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Olive">X</font> &amp; <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">X</font> )
 
;<br/>
<b>hence </b><a NAME="E3:76_1_1_2_3_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E14:76_1_1_2_3_1_1"/><b>then </b><i><font color="Green">E111</font></i>: 
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T19" target="_self">Th19</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E15:76_1_1_2_3_1_1"/><b>then </b><i><font color="Green">E112</font></i>: 
<font color="Maroon">Y</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/><a NAME="E16:76_1_1_2_3_1_1"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="xboole_0.html#K1">{}</a> ]
 ;<br/><a NAME="E17:76_1_1_2_3_1_1"/><i><font color="Green">E114</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Olive">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">B</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">B</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_1_2_3_1_1_3"><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">B</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:76_1_1_2_3_1_1_3"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E2:76_1_1_2_3_1_1_3"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E3:76_1_1_2_3_1_1_3"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">B</font>]
 <b>and </b><br/><a NAME="E4:76_1_1_2_3_1_1_3"/>
( not <font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> )
 ;<br/>

<a NAME="E5:76_1_1_2_3_1_1_3"/><i><font color="Green">E118</font></i>: 
 <a href="tarski.html#K3">union</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T31">ZFMISC_1:31</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_1_2_3_1_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:76_1_1_2_3_1_1_3_1"/>
( <font color="Maroon">B</font> is <a href="xboole_0.html#V1">empty</a> or  not <font color="Maroon">B</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="76_1_1_2_3_1_1_3_1_1"><b>suppose </b></a><a NAME="E1:76_1_1_2_3_1_1_3_1_1"/>
<font color="Maroon">B</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:76_1_1_2_3_1_1_3_1_1"/>
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T31" target="_self">Th31</a>, <a class="ref" href="zfmisc_1.html#T31">ZFMISC_1:31</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_1_2_3_1_1_3_1_2"><b>suppose </b></a><a NAME="E1:76_1_1_2_3_1_1_3_1_2"/>
not <font color="Maroon">B</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:76_1_1_2_3_1_1_3_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T32" target="_self">Th32</a>, <a class="ref" href="rcomp_3.html#T33" target="_self">Th33</a>, <a class="ref" href="rcomp_3.html#T31" target="_self">Th31</a>, <a class="ref" href="rcomp_3.html#T26" target="_self">Th26</a></i>;<br/><b>hence </b><a NAME="E3:76_1_1_2_3_1_1_3_1_2"/>
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T34" target="_self">Th34</a>, <a class="ref" href="zfmisc_1.html#T96">ZFMISC_1:96</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E18:76_1_1_2_3_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">Y</font>]
 <b>from </b><i><a class="ref" href="finset_1.html#S2">FINSET_1:sch 2</a>(<a class="ref" href="rcomp_3.html#T28" target="_self">Th28</a>, <a class="ref" href="rcomp_3.html#T29" target="_self">Th29</a>, <a class="ref" href="rcomp_3.html#T30" target="_self">Th30</a>);<br/></i><a NAME="E19:76_1_1_2_3_1_1"/><b>then </b><i><font color="Green">E119</font></i>: 
 <a href="tarski.html#K3">union</a> <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E22"><i><font color="Green">Lemma65</font></i></a>, <a class="ref" href="rcomp_3.html#T27" target="_self">Th27</a></i>;<br/><a NAME="E20:76_1_1_2_3_1_1"/>
<font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T24" target="_self">Th24</a>, <a class="ref" href="rcomp_3.html#T22" target="_self">Th22</a>, <a class="ref" href="rcomp_3.html#T23" target="_self">Th23</a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><b>hence </b><a NAME="E21:76_1_1_2_3_1_1"/><i><font color="Green">E120</font></i>: 
 <a href="tarski.html#K3">union</a> <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Olive">X</font> &amp; <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">X</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rcomp_3.html#T25" target="_self">Th25</a>, <a class="ref" href="rcomp_3.html#T31" target="_self">Th31</a></i>;<br/><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E22:76_1_1_2_3_1_1"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/><a NAME="E23:76_1_1_2_3_1_1"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">z</font>, <a href="tarski.html#K3">union</a> <font color="Maroon">Y</font>]
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/><b>hence </b><a NAME="E24:76_1_1_2_3_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,<span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">Y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T12" target="_self">Th12</a>, <a class="ref" href="rcomp_3.html#T32" target="_self">Th32</a>, <a class="ref" href="rcomp_3.html#T19" target="_self">Th19</a>, <a class="ref" href="rcomp_3.html#T33" target="_self">Th33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_1_2_3_1_2"><b>suppose </b></a><a NAME="E1:76_1_1_2_3_1_2"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">Y</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>take </b>
<font color="Maroon">w</font>
;<br/><b>thus </b><a NAME="E2:76_1_1_2_3_1_2"/>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Olive">X</font> &amp; <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">X</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rcomp_3.html#T7" target="_self">Th7</a></i>;<br/><b>thus </b><a NAME="E3:76_1_1_2_3_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">w</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 <b>by </b><i><a class="txt" href="rcomp_3.html#E22"><i><font color="Green">Lemma65</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>then consider </b><font color="Maroon">M</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E12:76_1_1_2"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">M</font> <a href="orders_1.html#R6">is_maximal_in</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T18" target="_self">Th18</a>, <a class="ref" href="rcomp_3.html#T16" target="_self">Th16</a>, <a class="ref" href="orders_1.html#T173">ORDERS_1:173</a></i>;<br/><a NAME="E13:76_1_1_2"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">M</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T19" target="_self">Th19</a>, <a class="ref" href="orders_1.html#D11">ORDERS_1:def 11</a></i>;<br/><a NAME="E14:76_1_1_2"/><b>then </b><i><font color="Green">E125</font></i>: 
 ex <font color="Olive">X</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> &amp; <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Olive">X</font> &amp; <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">X</font> )
 <b>by </b><i><a class="ref" href="rcomp_3.html#T16" target="_self">Th16</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_1_2_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:76_1_1_2_4"/>
<font color="Maroon">M</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">H</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:76_1_1_2_4"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font>
 <b>and </b><br/><a NAME="E4:76_1_1_2_4"/>
<font color="Maroon">H</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E5:76_1_1_2_4"/><i><font color="Green">E160</font></i>: 
 ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Maroon">H</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> )
 ;<br/><b>consider </b><font color="Maroon">Y</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E7:76_1_1_2_4"/><i><font color="Green">E161</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E8:76_1_1_2_4"/><i><font color="Green">E162</font></i>: 
<font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E9:76_1_1_2_4"/><i><font color="Green">E163</font></i>: 
<font color="Maroon">H</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T21" target="_self">Th21</a></i>;<br/><a NAME="E10:76_1_1_2_4"/><i><font color="Green">E164</font></i>: 
<font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E22"><i><font color="Green">Lemma65</font></i></a>, <a class="txt" href="rcomp_3.html#E23"><i><font color="Green">Lemma66</font></i></a>, <a class="ref" href="rcomp_3.html#T23" target="_self">Th23</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E11:76_1_1_2_4"/>
<font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E22"><i><font color="Green">Lemma65</font></i></a>, <a class="txt" href="rcomp_3.html#E23"><i><font color="Green">Lemma66</font></i></a>, <a class="ref" href="rcomp_3.html#T23" target="_self">Th23</a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><a NAME="E12:76_1_1_2_4"/><b>then </b><i><font color="Green">E165</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Olive">X</font> &amp; <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">X</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rcomp_3.html#T22" target="_self">Th22</a>, <a class="ref" href="rcomp_3.html#T25" target="_self">Th25</a></i>;<br/><a NAME="E13:76_1_1_2_4"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">Y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T12" target="_self">Th12</a>, <a class="ref" href="rcomp_3.html#T16" target="_self">Th16</a>, <a class="ref" href="rcomp_3.html#T20" target="_self">Th20</a>, <a class="ref" href="rcomp_3.html#T23" target="_self">Th23</a>, <a class="txt" href="rcomp_3.html#E23"><i><font color="Green">Lemma66</font></i></a></i>;<br/><b>hence </b><a NAME="E14:76_1_1_2_4"/>
contradiction
 <b>by </b><i><a class="ref" href="rcomp_3.html#T16" target="_self">Th16</a>, <a class="ref" href="rcomp_3.html#T24" target="_self">Th24</a>, <a class="txt" href="rcomp_3.html#E23"><i><font color="Green">Lemma66</font></i></a>, <a class="ref" href="rcomp_3.html#T26" target="_self">Th26</a>, <a class="ref" href="rcomp_3.html#T19" target="_self">Th19</a>, <a class="ref" href="orders_1.html#D11">ORDERS_1:def 11</a></i>;<br/></div><b>end;</b></div><a NAME="E16:76_1_1_2"/><b>then </b>
<font color="Maroon">M</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F1</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E22"><i><font color="Green">Lemma65</font></i></a>, <a class="ref" href="rcomp_3.html#T4" target="_self">Th4</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E17:76_1_1_2"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F1</font>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T5" target="_self">Th5</a>, <a class="txt" href="rcomp_3.html#E22"><i><font color="Green">Lemma65</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>thus </b><a NAME="E4:76"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">F1</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font>
 ;<br/>


</div>
