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

<b>let </b><font color="Maroon">R</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="relat_2.html#V4">antisymmetric</a> <a href="relat_2.html#V8">transitive</a> <a href="relat_1.html#NM1">Relation</a>;<br/><b>let </b><font color="Maroon">X</font> be  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:4"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>reconsider </b><font color="Maroon">IR</font> = <font color="Maroon">R</font> as   <a href="relset_1.html#NM3">Relation</a> of  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> <b>by </b><i><a class="ref" href="polynom1.html#T4">POLYNOM1:4</a></i>;<br/>
<b>set </b><font color="Maroon">S</font> =  <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">IR</font> #);<br/>
<b>set </b><font color="Maroon">CR</font> = the <a href="struct_0.html#U1">carrier</a> of <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">IR</font> #);<br/>
<b>set </b><font color="Maroon">ir</font> = the <a href="orders_2.html#U1">InternalRel</a> of <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">IR</font> #);<br/>
<a NAME="E3:4"/><i><font color="Green">E34</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">IR</font> #) is <a href="xboole_0.html#V1">(C1)</a>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>
<a NAME="E4:4"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">R</font> <a href="relat_2.html#R4">is_antisymmetric_in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="relat_2.html#D12">RELAT_2:def 12</a></i>;<br/>
<a NAME="E5:4"/><i><font color="Green">E38</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">IR</font> #) is <a href="orders_2.html#V4">antisymmetric</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">IR</font> #), <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">IR</font> #); <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D3">YELLOW_0:def 3</a><br/>

<b>assume </b><a NAME="E1:4_1"/>
( <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">x</font> )
 ;<br/>

<a NAME="E2:4_1"/><b>then </b>
( <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> the <a href="orders_2.html#U1">InternalRel</a> of <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">IR</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> the <a href="orders_2.html#U1">InternalRel</a> of <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">IR</font> #) )
 
<b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<b>hence </b><a NAME="E3:4_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i>, , <a class="ref" href="relat_2.html#D4">RELAT_2:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:4"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">R</font> <a href="relat_2.html#R8">is_transitive_in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="relat_2.html#D16">RELAT_2:def 16</a></i>;<br/>
<a NAME="E7:4"/>
 <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">IR</font> #) is <a href="orders_2.html#V3">transitive</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">IR</font> #), <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">IR</font> #), <font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">IR</font> #); <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a><br/>

<b>assume </b><a NAME="E1:4_2"/>
( <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">z</font> )
 ;<br/>

<a NAME="E2:4_2"/><b>then </b>
( <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> the <a href="orders_2.html#U1">InternalRel</a> of <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">IR</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> the <a href="orders_2.html#U1">InternalRel</a> of <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">IR</font> #) )
 
<b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E3:4_2"/><b>then </b>
<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> the <a href="orders_2.html#U1">InternalRel</a> of <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">IR</font> #)
 
<b>by </b><i>, , <a class="ref" href="relat_2.html#D8">RELAT_2:def 8</a></i>;<br/>
<b>hence </b><a NAME="E4:4_2"/>
<font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">S</font> =  <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">IR</font> #) as  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V3">transitive</a> <a href="orders_2.html#V4">antisymmetric</a>  <a href="orders_2.html#L1">RelStr</a>  <b>by </b><i>, , <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Y</font> = <font color="Maroon">X</font> as  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">IR</font> #) ;<br/>
<b>consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E11:4"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">x</font> <a href="waybel_4.html#R2">is_maximal_wrt</a> <font color="Maroon">Y</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> )
 <b>by </b><i>, <a class="ref" href="bagorder.html#T7">BAGORDER:7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">x</font>
;<br/>

<b>thus </b><a NAME="E13:4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_4.html#D24">WAYBEL_4:def 24</a><br/>

<b>given </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E14:4"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 ;<br/>

<b>thus </b><a NAME="E15:4"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="waybel_4.html#D24">WAYBEL_4:def 24</a></i>;<br/>


</div>
