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

<b>let </b><font color="Maroon">n</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>;<br/>



<b>set </b><font color="Maroon">R</font> =  <a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">n</font>;<br/>
<b>set </b><font color="Maroon">s</font> =  <a href="polynom2.html#K1" target="_self">support</a> <font color="Maroon">b</font>;<br/>
<a NAME="E1:29"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1" target="_self">support</a> <font color="Maroon">b</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1" target="_self">support</a> <font color="Maroon">b</font> &amp; <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font> &amp; not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">n</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:29_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1" target="_self">support</a> <font color="Maroon">b</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1" target="_self">support</a> <font color="Maroon">b</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> )
 ;<br/>

<b>assume </b><a NAME="E2:29_1"/>
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E3:29_1"/><b>then </b><i><font color="Green">E34</font></i>: 
not <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">y</font>
 
<b>by </b><i>, <a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font>, <font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<a NAME="E5:29_1"/>
<font color="Maroon">y</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">x</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:29_1"/>
<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>  <a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:29"/><b>then </b><i><font color="Green">E35</font></i>: 
 <a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">n</font> <a href="relat_2.html#R6">is_connected_in</a>  <a href="polynom2.html#K1" target="_self">support</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="relat_2.html#D6">RELAT_2:def 6</a></i>;<br/>
<a NAME="E3:29"/>
(  <a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">n</font> <a href="relat_2.html#R1">is_reflexive_in</a>  <a href="polynom2.html#K1" target="_self">support</a> <font color="Maroon">b</font> &amp;  <a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">n</font> <a href="relat_2.html#R4">is_antisymmetric_in</a>  <a href="polynom2.html#K1" target="_self">support</a> <font color="Maroon">b</font> &amp;  <a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">n</font> <a href="relat_2.html#R8">is_transitive_in</a>  <a href="polynom2.html#K1" target="_self">support</a> <font color="Maroon">b</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:29"/>
 <a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">n</font> <a href="orders_1.html#R3">linearly_orders</a>  <a href="polynom2.html#K1" target="_self">support</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="orders_1.html#D8">ORDERS_1:def 8</a></i>;<br/>


</div>
