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

<b>let </b><font color="Maroon" title="c1">R</font> be   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a>;<br/><b>let </b><font color="Maroon" title="c2">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><a NAME="E1:95"/><i><font color="Green" title="E53">A1</font></i>: 
( <font color="Maroon" title="c1">R</font> <a href="orders_1.html#R2" title="ORDERS_1:pred.2">partially_orders</a> <font color="Maroon" title="c2">X</font> &amp;  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X</font> )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">P</font> being  <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">P</font> &amp; <font color="Maroon" title="c1">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b1">P</font> <a href="orders_1.html#R2" title="ORDERS_1:pred.2">partially_orders</a> <font color="Maroon" title="c2">X</font> &amp;  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X</font> );<br/>
<b>consider </b><font color="Maroon" title="c3">Rel</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:95"/><i><font color="Green" title="E54">A2</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Rel</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c2">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1" title="XBOOLE_0:sch.1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E4:95"/>
<font color="Maroon" title="c1">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c2">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="orders_1.html#E1:95"><i><font color="Green" title="E53">A1</font></i></a>, <a class="txt" href="orders_1.html#E121"><i><font color="Green" title="E16">Lm4</font></i></a></i>;<br/>
<a NAME="E5:95"/><b>then </b><i><font color="Green" title="E55">A3</font></i>: 
<font color="Maroon" title="c3">Rel</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="orders_1.html#E1:95"><i><font color="Green" title="E53">A1</font></i></a>, <a class="txt" href="orders_1.html#E3:95"><i><font color="Green" title="E54">A2</font></i></a></i>;<br/>
<a NAME="E6:95"/>
for <font color="Olive" title="b1">Z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b1">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Rel</font> &amp; <font color="Olive" title="b1">Z</font> is <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a> holds <br/> <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Olive" title="b1">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Rel</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">Z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:95_1"/><i><font color="Green" title="E56">A4</font></i>: 
( <font color="Maroon" title="c4">Z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c4">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Rel</font> )
 <b>and </b><br/><a NAME="E2:95_1"/><i><font color="Green" title="E57">A5</font></i>: 
<font color="Maroon" title="c4">Z</font> is <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c5">T</font> = <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c2">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> as   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a> ;<br/>
<a NAME="E4:95_1"/>
<font color="Maroon" title="c4">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c2">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:95_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Z</font>
 ;<br/>

<b>hence </b><a NAME="E2:95_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c2">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="orders_1.html#E3:95"><i><font color="Green" title="E54">A2</font></i></a>, <a class="txt" href="orders_1.html#E1:95_1"><i><font color="Green" title="E56">A4</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:95_1"/><b>then </b><i><font color="Green" title="E58">A6</font></i>: 
(  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c4">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c2">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> &amp;  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c2">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">T</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T95" title="ZFMISC_1:th.95">ZFMISC_1:95</a>, <a class="ref" href="zfmisc_1.html#T99" title="ZFMISC_1:th.99">ZFMISC_1:99</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">S</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c4">Z</font> as   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a> <b>by </b><i><a class="ref" href="relat_1.html#T3" title="RELAT_1:th.3">RELAT_1:3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">Z</font>;<br/>
<a NAME="E8:95_1"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Rel</font>
 
<b>by </b><i><a class="txt" href="orders_1.html#E1:95_1"><i><font color="Green" title="E56">A4</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">P</font> being   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a><b> such that </b><br/><a NAME="E10:95_1"/><i><font color="Green" title="E59">A7</font></i>: 
( <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">P</font> &amp; <font color="Maroon" title="c1">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">P</font> &amp; <font color="Maroon" title="c8">P</font> <a href="orders_1.html#R2" title="ORDERS_1:pred.2">partially_orders</a> <font color="Maroon" title="c2">X</font> &amp;  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c8">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X</font> )
 <b>by </b><i><a class="txt" href="orders_1.html#E3:95"><i><font color="Green" title="E54">A2</font></i></a></i>;<br/>
<a NAME="E11:95_1"/><i><font color="Green" title="E60">A8</font></i>: 
<font color="Maroon" title="c8">P</font> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a> <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="txt" href="orders_1.html#E10:95_1"><i><font color="Green" title="E59">A7</font></i></a>, <a class="ref" href="orders_1.html#D7" target="_self" title="ORDERS_1:def.7">Def7</a></i>;<br/>
<a NAME="E12:95_1"/><i><font color="Green" title="E61">A9</font></i>: 
<font color="Maroon" title="c1">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_1.html#D3" title="RELAT_1:def.3">RELAT_1:def 3</a><br/><b>let </b><font color="Maroon" title="c10">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><a NAME="E1:95_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font>
 ;<br/>

<b>hence </b><a NAME="E2:95_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S</font>
 <b>by </b><i><a class="txt" href="orders_1.html#E1:95_1"><i><font color="Green" title="E56">A4</font></i></a>, <a class="txt" href="orders_1.html#E10:95_1"><i><font color="Green" title="E59">A7</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:95_1"/><i><font color="Green" title="E62">A10</font></i>: 
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:95_1_3"/>
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:95_1_3_1"/><i><font color="Green" title="E63">A11</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">S</font>
 ;<br/>

<div><i><font color="Green" title="E64">A12</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_3_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:95_1_3_1_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">S</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c10">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:95_1_3_1_1"/><i><font color="Green" title="E65">A13</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D4" title="RELAT_1:def.4">RELAT_1:def 4</a></i>;<br/><b>consider </b><font color="Maroon" title="c11">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:95_1_3_1_1"/><i><font color="Green" title="E66">A14</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Y</font> &amp; <font color="Maroon" title="c11">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Z</font> )
 <b>by </b><i><a class="txt" href="orders_1.html#E3:95_1_3_1_1"><i><font color="Green" title="E65">A13</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><a NAME="E6:95_1_3_1_1"/>
 ex <font color="Olive" title="b1">P</font> being  <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a> st <br/>( <font color="Maroon" title="c11">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">P</font> &amp; <font color="Maroon" title="c1">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b1">P</font> <a href="orders_1.html#R2" title="ORDERS_1:pred.2">partially_orders</a> <font color="Maroon" title="c2">X</font> &amp;  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X</font> )
 <b>by </b><i><a class="txt" href="orders_1.html#E3:95"><i><font color="Green" title="E54">A2</font></i></a>, <a class="txt" href="orders_1.html#E1:95_1"><i><font color="Green" title="E56">A4</font></i></a>, <a class="txt" href="orders_1.html#E5:95_1_3_1_1"><i><font color="Green" title="E66">A14</font></i></a></i>;<br/><b>hence </b><a NAME="E7:95_1_3_1_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <b>by </b><i><a class="txt" href="orders_1.html#E5:95_1_3_1_1"><i><font color="Green" title="E66">A14</font></i></a>, <a class="ref" href="relat_1.html#T30" title="RELAT_1:th.30">RELAT_1:30</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_3_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:95_1_3_1_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK4" title="RELAT_1:NK.4">rng</a> <font color="Maroon" title="c6">S</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c10">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:95_1_3_1_2"/><i><font color="Green" title="E65">A15</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c9">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D5" title="RELAT_1:def.5">RELAT_1:def 5</a></i>;<br/><b>consider </b><font color="Maroon" title="c11">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:95_1_3_1_2"/><i><font color="Green" title="E66">A16</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c9">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Y</font> &amp; <font color="Maroon" title="c11">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Z</font> )
 <b>by </b><i><a class="txt" href="orders_1.html#E3:95_1_3_1_2"><i><font color="Green" title="E65">A15</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><a NAME="E6:95_1_3_1_2"/>
 ex <font color="Olive" title="b1">P</font> being  <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a> st <br/>( <font color="Maroon" title="c11">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">P</font> &amp; <font color="Maroon" title="c1">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b1">P</font> <a href="orders_1.html#R2" title="ORDERS_1:pred.2">partially_orders</a> <font color="Maroon" title="c2">X</font> &amp;  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X</font> )
 <b>by </b><i><a class="txt" href="orders_1.html#E3:95"><i><font color="Green" title="E54">A2</font></i></a>, <a class="txt" href="orders_1.html#E1:95_1"><i><font color="Green" title="E56">A4</font></i></a>, <a class="txt" href="orders_1.html#E5:95_1_3_1_2"><i><font color="Green" title="E66">A16</font></i></a></i>;<br/><b>hence </b><a NAME="E7:95_1_3_1_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <b>by </b><i><a class="txt" href="orders_1.html#E5:95_1_3_1_2"><i><font color="Green" title="E66">A16</font></i></a>, <a class="ref" href="relat_1.html#T30" title="RELAT_1:th.30">RELAT_1:30</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:95_1_3_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <b>by </b><i><a class="txt" href="orders_1.html#E1:95_1_3_1"><i><font color="Green" title="E63">A11</font></i></a>, <a class="txt" href="orders_1.html#E2:95_1_3_1"><i><font color="Green" title="E64">A12</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>


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

<b>thus </b><a NAME="E2:95_1_3"/>
<font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">S</font>
 <b>by </b><i><a class="txt" href="orders_1.html#E1:95"><i><font color="Green" title="E53">A1</font></i></a>, <a class="txt" href="orders_1.html#E12:95_1"><i><font color="Green" title="E61">A9</font></i></a>, <a class="ref" href="relat_1.html#T31" title="RELAT_1:th.31">RELAT_1:31</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:95_1"/>
<font color="Maroon" title="c6">S</font> <a href="orders_1.html#R2" title="ORDERS_1:pred.2">partially_orders</a> <font color="Maroon" title="c2">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:95_1_4"/>
<font color="Maroon" title="c6">S</font> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a> <font color="Maroon" title="c2">X</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="orders_1.html#D7" target="_self" title="ORDERS_1:def.7">ORDERS_1:def 7</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_2.html#D1" title="RELAT_2:def.1">RELAT_2:def 1</a><br/>

<b>assume </b><a NAME="E1:95_1_4_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 ;<br/>

<a NAME="E2:95_1_4_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c9">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">P</font>
 
<b>by </b><i><a class="txt" href="orders_1.html#E11:95_1"><i><font color="Green" title="E60">A8</font></i></a>, <a class="ref" href="relat_2.html#D1" title="RELAT_2:def.1">RELAT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:95_1_4_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c9">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S</font>
 <b>by </b><i><a class="txt" href="orders_1.html#E1:95_1"><i><font color="Green" title="E56">A4</font></i></a>, <a class="txt" href="orders_1.html#E10:95_1"><i><font color="Green" title="E59">A7</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>


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

<b>thus </b><a NAME="E2:95_1_4"/>
<font color="Maroon" title="c6">S</font> <a href="relat_2.html#R8" title="RELAT_2:pred.8">is_transitive_in</a> <font color="Maroon" title="c2">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_2.html#D8" title="RELAT_2:def.8">RELAT_2:def 8</a><br/><b>let </b><font color="Maroon" title="c10">y</font>, <font color="Maroon" title="c11">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>





<b>assume </b><a NAME="E1:95_1_4_2"/><i><font color="Green" title="E63">A17</font></i>: 
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c11">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c11">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S</font> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">X1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:95_1_4_2"/><i><font color="Green" title="E64">A18</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">X1</font> &amp; <font color="Maroon" title="c12">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Z</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">X2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:95_1_4_2"/><i><font color="Green" title="E65">A19</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c11">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">X2</font> &amp; <font color="Maroon" title="c13">X2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Z</font> )
 <b>by </b><i><a class="txt" href="orders_1.html#E1:95_1_4_2"><i><font color="Green" title="E63">A17</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E6:95_1_4_2"/>
<font color="Maroon" title="c12">X1</font>,<font color="Maroon" title="c13">X2</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a> 
 
<b>by </b><i><a class="txt" href="orders_1.html#E2:95_1"><i><font color="Green" title="E57">A5</font></i></a>, <a class="txt" href="orders_1.html#E3:95_1_4_2"><i><font color="Green" title="E64">A18</font></i></a>, <a class="txt" href="orders_1.html#E5:95_1_4_2"><i><font color="Green" title="E65">A19</font></i></a>, <a class="ref" href="ordinal1.html#D9" title="ORDINAL1:def.9">ORDINAL1:def 9</a></i>;<br/>
<a NAME="E7:95_1_4_2"/><b>then </b><i><font color="Green" title="E66">A20</font></i>: 
( <font color="Maroon" title="c12">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">X2</font> or <font color="Maroon" title="c13">X2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">X1</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D9" title="XBOOLE_0:def.9">XBOOLE_0:def 9</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">P1</font> being   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a><b> such that </b><br/><a NAME="E9:95_1_4_2"/><i><font color="Green" title="E67">A21</font></i>: 
( <font color="Maroon" title="c12">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">P1</font> &amp; <font color="Maroon" title="c1">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">P1</font> &amp; <font color="Maroon" title="c14">P1</font> <a href="orders_1.html#R2" title="ORDERS_1:pred.2">partially_orders</a> <font color="Maroon" title="c2">X</font> &amp;  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c14">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X</font> )
 <b>by </b><i><a class="txt" href="orders_1.html#E3:95"><i><font color="Green" title="E54">A2</font></i></a>, <a class="txt" href="orders_1.html#E1:95_1"><i><font color="Green" title="E56">A4</font></i></a>, <a class="txt" href="orders_1.html#E3:95_1_4_2"><i><font color="Green" title="E64">A18</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c15">P2</font> being   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a><b> such that </b><br/><a NAME="E11:95_1_4_2"/><i><font color="Green" title="E68">A22</font></i>: 
( <font color="Maroon" title="c13">X2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">P2</font> &amp; <font color="Maroon" title="c1">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">P2</font> &amp; <font color="Maroon" title="c15">P2</font> <a href="orders_1.html#R2" title="ORDERS_1:pred.2">partially_orders</a> <font color="Maroon" title="c2">X</font> &amp;  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c15">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X</font> )
 <b>by </b><i><a class="txt" href="orders_1.html#E3:95"><i><font color="Green" title="E54">A2</font></i></a>, <a class="txt" href="orders_1.html#E1:95_1"><i><font color="Green" title="E56">A4</font></i></a>, <a class="txt" href="orders_1.html#E5:95_1_4_2"><i><font color="Green" title="E65">A19</font></i></a></i>;<br/>
<a NAME="E12:95_1_4_2"/>
( <font color="Maroon" title="c14">P1</font> <a href="relat_2.html#R8" title="RELAT_2:pred.8">is_transitive_in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c15">P2</font> <a href="relat_2.html#R8" title="RELAT_2:pred.8">is_transitive_in</a> <font color="Maroon" title="c2">X</font> )
 
<b>by </b><i><a class="txt" href="orders_1.html#E9:95_1_4_2"><i><font color="Green" title="E67">A21</font></i></a>, <a class="txt" href="orders_1.html#E11:95_1_4_2"><i><font color="Green" title="E68">A22</font></i></a>, <a class="ref" href="orders_1.html#D7" target="_self" title="ORDERS_1:def.7">Def7</a></i>;<br/>
<a NAME="E13:95_1_4_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c11">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">P1</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c11">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">P2</font> )
 
<b>by </b><i><a class="txt" href="orders_1.html#E1:95_1_4_2"><i><font color="Green" title="E63">A17</font></i></a>, <a class="txt" href="orders_1.html#E3:95_1_4_2"><i><font color="Green" title="E64">A18</font></i></a>, <a class="txt" href="orders_1.html#E5:95_1_4_2"><i><font color="Green" title="E65">A19</font></i></a>, <a class="txt" href="orders_1.html#E7:95_1_4_2"><i><font color="Green" title="E66">A20</font></i></a>, <a class="txt" href="orders_1.html#E9:95_1_4_2"><i><font color="Green" title="E67">A21</font></i></a>, <a class="txt" href="orders_1.html#E11:95_1_4_2"><i><font color="Green" title="E68">A22</font></i></a>, <a class="ref" href="relat_2.html#D8" title="RELAT_2:def.8">RELAT_2:def 8</a></i>;<br/>
<b>hence </b><a NAME="E14:95_1_4_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c11">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S</font>
 <b>by </b><i><a class="txt" href="orders_1.html#E3:95_1_4_2"><i><font color="Green" title="E64">A18</font></i></a>, <a class="txt" href="orders_1.html#E5:95_1_4_2"><i><font color="Green" title="E65">A19</font></i></a>, <a class="txt" href="orders_1.html#E9:95_1_4_2"><i><font color="Green" title="E67">A21</font></i></a>, <a class="txt" href="orders_1.html#E11:95_1_4_2"><i><font color="Green" title="E68">A22</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>


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

<b>thus </b><a NAME="E3:95_1_4"/>
<font color="Maroon" title="c6">S</font> <a href="relat_2.html#R4" title="RELAT_2:pred.4">is_antisymmetric_in</a> <font color="Maroon" title="c2">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_4_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_2.html#D4" title="RELAT_2:def.4">RELAT_2:def 4</a><br/><b>let </b><font color="Maroon" title="c10">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><a NAME="E1:95_1_4_3"/><i><font color="Green" title="E63">A23</font></i>: 
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c9">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S</font> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c11">X1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:95_1_4_3"/><i><font color="Green" title="E64">A24</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">X1</font> &amp; <font color="Maroon" title="c11">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Z</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">X2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:95_1_4_3"/><i><font color="Green" title="E65">A25</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c9">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">X2</font> &amp; <font color="Maroon" title="c12">X2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Z</font> )
 <b>by </b><i><a class="txt" href="orders_1.html#E1:95_1_4_3"><i><font color="Green" title="E63">A23</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E6:95_1_4_3"/>
<font color="Maroon" title="c11">X1</font>,<font color="Maroon" title="c12">X2</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a> 
 
<b>by </b><i><a class="txt" href="orders_1.html#E2:95_1"><i><font color="Green" title="E57">A5</font></i></a>, <a class="txt" href="orders_1.html#E3:95_1_4_3"><i><font color="Green" title="E64">A24</font></i></a>, <a class="txt" href="orders_1.html#E5:95_1_4_3"><i><font color="Green" title="E65">A25</font></i></a>, <a class="ref" href="ordinal1.html#D9" title="ORDINAL1:def.9">ORDINAL1:def 9</a></i>;<br/>
<a NAME="E7:95_1_4_3"/><b>then </b><i><font color="Green" title="E66">A26</font></i>: 
( <font color="Maroon" title="c11">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">X2</font> or <font color="Maroon" title="c12">X2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">X1</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D9" title="XBOOLE_0:def.9">XBOOLE_0:def 9</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">P1</font> being   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a><b> such that </b><br/><a NAME="E9:95_1_4_3"/><i><font color="Green" title="E67">A27</font></i>: 
( <font color="Maroon" title="c11">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">P1</font> &amp; <font color="Maroon" title="c1">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">P1</font> &amp; <font color="Maroon" title="c13">P1</font> <a href="orders_1.html#R2" title="ORDERS_1:pred.2">partially_orders</a> <font color="Maroon" title="c2">X</font> &amp;  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c13">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X</font> )
 <b>by </b><i><a class="txt" href="orders_1.html#E3:95"><i><font color="Green" title="E54">A2</font></i></a>, <a class="txt" href="orders_1.html#E1:95_1"><i><font color="Green" title="E56">A4</font></i></a>, <a class="txt" href="orders_1.html#E3:95_1_4_3"><i><font color="Green" title="E64">A24</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">P2</font> being   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a><b> such that </b><br/><a NAME="E11:95_1_4_3"/><i><font color="Green" title="E68">A28</font></i>: 
( <font color="Maroon" title="c12">X2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">P2</font> &amp; <font color="Maroon" title="c1">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">P2</font> &amp; <font color="Maroon" title="c14">P2</font> <a href="orders_1.html#R2" title="ORDERS_1:pred.2">partially_orders</a> <font color="Maroon" title="c2">X</font> &amp;  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c14">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X</font> )
 <b>by </b><i><a class="txt" href="orders_1.html#E3:95"><i><font color="Green" title="E54">A2</font></i></a>, <a class="txt" href="orders_1.html#E1:95_1"><i><font color="Green" title="E56">A4</font></i></a>, <a class="txt" href="orders_1.html#E5:95_1_4_3"><i><font color="Green" title="E65">A25</font></i></a></i>;<br/>
<a NAME="E12:95_1_4_3"/>
( <font color="Maroon" title="c13">P1</font> <a href="relat_2.html#R4" title="RELAT_2:pred.4">is_antisymmetric_in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c14">P2</font> <a href="relat_2.html#R4" title="RELAT_2:pred.4">is_antisymmetric_in</a> <font color="Maroon" title="c2">X</font> )
 
<b>by </b><i><a class="txt" href="orders_1.html#E9:95_1_4_3"><i><font color="Green" title="E67">A27</font></i></a>, <a class="txt" href="orders_1.html#E11:95_1_4_3"><i><font color="Green" title="E68">A28</font></i></a>, <a class="ref" href="orders_1.html#D7" target="_self" title="ORDERS_1:def.7">Def7</a></i>;<br/>
<b>hence </b><a NAME="E13:95_1_4_3"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y</font>
 <b>by </b><i><a class="txt" href="orders_1.html#E1:95_1_4_3"><i><font color="Green" title="E63">A23</font></i></a>, <a class="txt" href="orders_1.html#E3:95_1_4_3"><i><font color="Green" title="E64">A24</font></i></a>, <a class="txt" href="orders_1.html#E5:95_1_4_3"><i><font color="Green" title="E65">A25</font></i></a>, <a class="txt" href="orders_1.html#E7:95_1_4_3"><i><font color="Green" title="E66">A26</font></i></a>, <a class="txt" href="orders_1.html#E9:95_1_4_3"><i><font color="Green" title="E67">A27</font></i></a>, <a class="txt" href="orders_1.html#E11:95_1_4_3"><i><font color="Green" title="E68">A28</font></i></a>, <a class="ref" href="relat_2.html#D4" title="RELAT_2:def.4">RELAT_2:def 4</a></i>;<br/>


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E15:95_1"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c4">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Rel</font>
 <b>by </b><i><a class="txt" href="orders_1.html#E3:95"><i><font color="Green" title="E54">A2</font></i></a>, <a class="txt" href="orders_1.html#E5:95_1"><i><font color="Green" title="E58">A6</font></i></a>, <a class="txt" href="orders_1.html#E12:95_1"><i><font color="Green" title="E61">A9</font></i></a>, <a class="txt" href="orders_1.html#E13:95_1"><i><font color="Green" title="E62">A10</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c4">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:95"/><i><font color="Green" title="E56">A29</font></i>: 
( <font color="Maroon" title="c4">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Rel</font> &amp; ( for <font color="Olive" title="b1">Z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Rel</font> &amp; <font color="Olive" title="b1">Z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">Y</font> holds <br/> not <font color="Maroon" title="c4">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Z</font> ) )
 <b>by </b><i><a class="txt" href="orders_1.html#E5:95"><i><font color="Green" title="E55">A3</font></i></a>, <a class="ref" href="orders_1.html#T177" target="_self" title="ORDERS_1:th.177">Th177</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">P</font> being   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a><b> such that </b><br/><a NAME="E10:95"/><i><font color="Green" title="E57">A30</font></i>: 
( <font color="Maroon" title="c4">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Maroon" title="c1">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Maroon" title="c5">P</font> <a href="orders_1.html#R2" title="ORDERS_1:pred.2">partially_orders</a> <font color="Maroon" title="c2">X</font> &amp;  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X</font> )
 <b>by </b><i><a class="txt" href="orders_1.html#E3:95"><i><font color="Green" title="E54">A2</font></i></a>, <a class="txt" href="orders_1.html#E8:95"><i><font color="Green" title="E56">A29</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">P</font>
;<br/>

<b>thus </b><a NAME="E11:95"/>
<font color="Maroon" title="c1">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="orders_1.html#E10:95"><i><font color="Green" title="E57">A30</font></i></a></i>;<br/>

<b>thus </b><a NAME="E12:95"/><i><font color="Green" title="E58">A31</font></i>: 
( <font color="Maroon" title="c5">P</font> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c5">P</font> <a href="relat_2.html#R8" title="RELAT_2:pred.8">is_transitive_in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c5">P</font> <a href="relat_2.html#R4" title="RELAT_2:pred.4">is_antisymmetric_in</a> <font color="Maroon" title="c2">X</font> )
 <b>by </b><i><a class="txt" href="orders_1.html#E10:95"><i><font color="Green" title="E57">A30</font></i></a>, <a class="ref" href="orders_1.html#D7" target="_self" title="ORDERS_1:def.7">Def7</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="orders_1.html#D8" target="_self" title="ORDERS_1:def.8">ORDERS_1:def 8</a><br/>

<b>thus </b><a NAME="E13:95"/>
<font color="Maroon" title="c5">P</font> <a href="relat_2.html#R6" title="RELAT_2:pred.6">is_connected_in</a> <font color="Maroon" title="c2">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_2.html#D6" title="RELAT_2:def.6">RELAT_2:def 6</a><br/><b>let </b><font color="Maroon" title="c7">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><a NAME="E1:95_2"/><i><font color="Green" title="E59">A32</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">y</font> &amp; not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> &amp; not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c6">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">$1,$2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> or ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">$1,<font color="Maroon" title="c6">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">y</font>,$2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> ) );<br/>
<b>consider </b><font color="Maroon" title="c8">Q</font> being   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a><b> such that </b><br/><a NAME="E3:95_2"/><i><font color="Green" title="E60">A33</font></i>: 
for <font color="Olive" title="b1">z</font>, <font color="Olive" title="b2">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">z</font>,<font color="Olive" title="b2">u</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Q</font> iff ( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">z</font>,<font color="Olive" title="b2">u</font>] ) )
 <b>from </b><i><a class="ref" href="relat_1.html#S1" title="RELAT_1:sch.1">RELAT_1:sch 1</a>();<br/></i>
<a NAME="E4:95_2"/><i><font color="Green" title="E61">A34</font></i>: 
<font color="Maroon" title="c5">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">Q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_1.html#D3" title="RELAT_1:def.3">RELAT_1:def 3</a><br/><b>let </b><font color="Maroon" title="c10">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><a NAME="E1:95_2_1"/><i><font color="Green" title="E62">A35</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">z</font>,<font color="Maroon" title="c10">u</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 ;<br/>

<a NAME="E2:95_2_1"/><b>then </b>
( <font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c5">P</font> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T30" title="RELAT_1:th.30">RELAT_1:30</a></i>;<br/>
<b>hence </b><a NAME="E3:95_2_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">z</font>,<font color="Maroon" title="c10">u</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Q</font>
 <b>by </b><i><a class="txt" href="orders_1.html#E10:95"><i><font color="Green" title="E57">A30</font></i></a>, <a class="txt" href="orders_1.html#E3:95_2"><i><font color="Green" title="E60">A33</font></i></a>, <a class="txt" href="orders_1.html#E1:95_2_1"><i><font color="Green" title="E62">A35</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:95_2"/><b>then </b><i><font color="Green" title="E62">A36</font></i>: 
<font color="Maroon" title="c1">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">Q</font>
 
<b>by </b><i><a class="txt" href="orders_1.html#E10:95"><i><font color="Green" title="E57">A30</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E6:95_2"/><i><font color="Green" title="E63">A37</font></i>: 
<font color="Maroon" title="c8">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c2">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_1.html#D3" title="RELAT_1:def.3">RELAT_1:def 3</a><br/><b>let </b><font color="Maroon" title="c10">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><a NAME="E1:95_2_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c10">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Q</font>
 ;<br/>

<a NAME="E2:95_2_2"/><b>then </b>
( <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c10">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> )
 
<b>by </b><i><a class="txt" href="orders_1.html#E3:95_2"><i><font color="Green" title="E60">A33</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:95_2_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c10">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c2">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:95_2"/><i><font color="Green" title="E64">A38</font></i>: 
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c8">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_2_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:95_2_3"/>
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c8">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_2_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:95_2_3_1"/><i><font color="Green" title="E65">A39</font></i>: 
<font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c8">Q</font>
 ;<br/>

<div><i><font color="Green" title="E66">A40</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="95_2_3_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:95_2_3_1_1"/>
<font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">Q</font>
 ;<br/><a NAME="E2:95_2_3_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">z</font>,<font color="Olive" title="b1">u</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Q</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D4" title="RELAT_1:def.4">RELAT_1:def 4</a></i>;<br/><b>hence </b><a NAME="E3:95_2_3_1_1"/>
<font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <b>by </b><i><a class="txt" href="orders_1.html#E3:95_2"><i><font color="Green" title="E60">A33</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_2_3_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:95_2_3_1_2"/>
<font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK4" title="RELAT_1:NK.4">rng</a> <font color="Maroon" title="c8">Q</font>
 ;<br/><a NAME="E2:95_2_3_1_2"/><b>then </b>
 ex <font color="Olive" title="b1">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">u</font>,<font color="Maroon" title="c9">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Q</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D5" title="RELAT_1:def.5">RELAT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:95_2_3_1_2"/>
<font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <b>by </b><i><a class="txt" href="orders_1.html#E3:95_2"><i><font color="Green" title="E60">A33</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:95_2_3_1"/>
<font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <b>by </b><i><a class="txt" href="orders_1.html#E1:95_2_3_1"><i><font color="Green" title="E65">A39</font></i></a>, <a class="txt" href="orders_1.html#E2:95_2_3_1"><i><font color="Green" title="E66">A40</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>


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

<b>thus </b><a NAME="E2:95_2_3"/>
<font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c8">Q</font>
 <b>by </b><i><a class="txt" href="orders_1.html#E10:95"><i><font color="Green" title="E57">A30</font></i></a>, <a class="txt" href="orders_1.html#E4:95_2"><i><font color="Green" title="E61">A34</font></i></a>, <a class="ref" href="relat_1.html#T31" title="RELAT_1:th.31">RELAT_1:31</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:95_2"/>
<font color="Maroon" title="c8">Q</font> <a href="orders_1.html#R2" title="ORDERS_1:pred.2">partially_orders</a> <font color="Maroon" title="c2">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_2_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:95_2_4"/>
<font color="Maroon" title="c8">Q</font> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a> <font color="Maroon" title="c2">X</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="orders_1.html#D7" target="_self" title="ORDERS_1:def.7">ORDERS_1:def 7</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_2_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_2.html#D1" title="RELAT_2:def.1">RELAT_2:def 1</a><br/>

<b>assume </b><a NAME="E1:95_2_4_1"/><i><font color="Green" title="E65">A41</font></i>: 
<font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 ;<br/>

<a NAME="E2:95_2_4_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">z</font>,<font color="Maroon" title="c9">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 
<b>by </b><i><a class="txt" href="orders_1.html#E12:95"><i><font color="Green" title="E58">A31</font></i></a>, <a class="ref" href="relat_2.html#D1" title="RELAT_2:def.1">RELAT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:95_2_4_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">z</font>,<font color="Maroon" title="c9">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Q</font>
 <b>by </b><i><a class="txt" href="orders_1.html#E3:95_2"><i><font color="Green" title="E60">A33</font></i></a>, <a class="txt" href="orders_1.html#E1:95_2_4_1"><i><font color="Green" title="E65">A41</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E2:95_2_4"/>
<font color="Maroon" title="c8">Q</font> <a href="relat_2.html#R8" title="RELAT_2:pred.8">is_transitive_in</a> <font color="Maroon" title="c2">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_2_4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">a</font>, <font color="Maroon" title="c10">b</font>, <font color="Maroon" title="c11">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_2.html#D8" title="RELAT_2:def.8">RELAT_2:def 8</a><br/>

<b>assume </b><a NAME="E1:95_2_4_2"/><i><font color="Green" title="E65">A42</font></i>: 
( <font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c10">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c11">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Q</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">b</font>,<font color="Maroon" title="c11">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Q</font> )
 ;<br/>

<a NAME="E2:95_2_4_2"/><b>then </b>
( ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> or ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c6">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c10">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> ) ) &amp; ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">b</font>,<font color="Maroon" title="c11">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> or ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">b</font>,<font color="Maroon" title="c6">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c11">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> ) ) )
 
<b>by </b><i><a class="txt" href="orders_1.html#E3:95_2"><i><font color="Green" title="E60">A33</font></i></a></i>;<br/>
<a NAME="E3:95_2_4_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c11">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> or ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c6">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c11">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> ) or ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c6">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c11">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> ) or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c6">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> )
 
<b>by </b><i><a class="txt" href="orders_1.html#E12:95"><i><font color="Green" title="E58">A31</font></i></a>, <a class="txt" href="orders_1.html#E1:95_2"><i><font color="Green" title="E59">A32</font></i></a>, <a class="txt" href="orders_1.html#E1:95_2_4_2"><i><font color="Green" title="E65">A42</font></i></a>, <a class="ref" href="relat_2.html#D8" title="RELAT_2:def.8">RELAT_2:def 8</a></i>;<br/>
<b>hence </b><a NAME="E4:95_2_4_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c11">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Q</font>
 <b>by </b><i><a class="txt" href="orders_1.html#E1:95_2"><i><font color="Green" title="E59">A32</font></i></a>, <a class="txt" href="orders_1.html#E3:95_2"><i><font color="Green" title="E60">A33</font></i></a>, <a class="txt" href="orders_1.html#E1:95_2_4_2"><i><font color="Green" title="E65">A42</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E3:95_2_4"/>
<font color="Maroon" title="c8">Q</font> <a href="relat_2.html#R4" title="RELAT_2:pred.4">is_antisymmetric_in</a> <font color="Maroon" title="c2">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_2_4_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">a</font>, <font color="Maroon" title="c10">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_2.html#D4" title="RELAT_2:def.4">RELAT_2:def 4</a><br/>

<b>assume </b><a NAME="E1:95_2_4_3"/><i><font color="Green" title="E65">A43</font></i>: 
( <font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c10">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Q</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">b</font>,<font color="Maroon" title="c9">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Q</font> )
 ;<br/>

<a NAME="E2:95_2_4_3"/><b>then </b>
( ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> or ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c6">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c10">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> ) ) &amp; ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">b</font>,<font color="Maroon" title="c9">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> or ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">b</font>,<font color="Maroon" title="c6">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c9">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> ) ) )
 
<b>by </b><i><a class="txt" href="orders_1.html#E3:95_2"><i><font color="Green" title="E60">A33</font></i></a></i>;<br/>
<a NAME="E3:95_2_4_3"/><b>then </b>
( <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font> or ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c6">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c9">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> ) or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c6">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> )
 
<b>by </b><i><a class="txt" href="orders_1.html#E12:95"><i><font color="Green" title="E58">A31</font></i></a>, <a class="txt" href="orders_1.html#E1:95_2"><i><font color="Green" title="E59">A32</font></i></a>, <a class="txt" href="orders_1.html#E1:95_2_4_3"><i><font color="Green" title="E65">A43</font></i></a>, <a class="ref" href="relat_2.html#D4" title="RELAT_2:def.4">RELAT_2:def 4</a>, <a class="ref" href="relat_2.html#D8" title="RELAT_2:def.8">RELAT_2:def 8</a></i>;<br/>
<b>hence </b><a NAME="E4:95_2_4_3"/>
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font>
 <b>by </b><i><a class="txt" href="orders_1.html#E12:95"><i><font color="Green" title="E58">A31</font></i></a>, <a class="txt" href="orders_1.html#E1:95_2"><i><font color="Green" title="E59">A32</font></i></a>, <a class="txt" href="orders_1.html#E1:95_2_4_3"><i><font color="Green" title="E65">A43</font></i></a>, <a class="ref" href="relat_2.html#D8" title="RELAT_2:def.8">RELAT_2:def 8</a></i>;<br/>


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


</div><b>end;</b></div>
<a NAME="E9:95_2"/><b>then </b>
<font color="Maroon" title="c8">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Rel</font>
 
<b>by </b><i><a class="txt" href="orders_1.html#E3:95"><i><font color="Green" title="E54">A2</font></i></a>, <a class="txt" href="orders_1.html#E5:95_2"><i><font color="Green" title="E62">A36</font></i></a>, <a class="txt" href="orders_1.html#E6:95_2"><i><font color="Green" title="E63">A37</font></i></a>, <a class="txt" href="orders_1.html#E7:95_2"><i><font color="Green" title="E64">A38</font></i></a></i>;<br/>
<a NAME="E10:95_2"/><b>then </b><i><font color="Green" title="E65">A44</font></i>: 
<font color="Maroon" title="c8">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font>
 
<b>by </b><i><a class="txt" href="orders_1.html#E8:95"><i><font color="Green" title="E56">A29</font></i></a>, <a class="txt" href="orders_1.html#E10:95"><i><font color="Green" title="E57">A30</font></i></a>, <a class="txt" href="orders_1.html#E4:95_2"><i><font color="Green" title="E61">A34</font></i></a></i>;<br/>
<a NAME="E11:95_2"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c6">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c7">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> )
 
<b>by </b><i><a class="txt" href="orders_1.html#E12:95"><i><font color="Green" title="E58">A31</font></i></a>, <a class="txt" href="orders_1.html#E1:95_2"><i><font color="Green" title="E59">A32</font></i></a>, <a class="ref" href="relat_2.html#D1" title="RELAT_2:def.1">RELAT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E12:95_2"/>
contradiction
 <b>by </b><i><a class="txt" href="orders_1.html#E1:95_2"><i><font color="Green" title="E59">A32</font></i></a>, <a class="txt" href="orders_1.html#E3:95_2"><i><font color="Green" title="E60">A33</font></i></a>, <a class="txt" href="orders_1.html#E10:95_2"><i><font color="Green" title="E65">A44</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E14:95"/>
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X</font>
 <b>by </b><i><a class="txt" href="orders_1.html#E10:95"><i><font color="Green" title="E57">A30</font></i></a></i>;<br/>


</div>
