<?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>set </b><font color="Maroon">BO</font> =  <a href="polynom1.html#K17" target="_self">BagOrder</a> <font color="Maroon">n</font>;<br/>
<a NAME="E1:133"/><i><font color="Green">E41</font></i>: 
 <a href="polynom1.html#K17" target="_self">BagOrder</a> <font color="Maroon">n</font> <a href="relat_2.html#R1">is_reflexive_in</a>  <a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:133"/><i><font color="Green">E42</font></i>: 
 <a href="polynom1.html#K17" target="_self">BagOrder</a> <font color="Maroon">n</font> <a href="relat_2.html#R4">is_antisymmetric_in</a>  <a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:133"/><i><font color="Green">E43</font></i>: 
 <a href="polynom1.html#K17" target="_self">BagOrder</a> <font color="Maroon">n</font> <a href="relat_2.html#R8">is_transitive_in</a>  <a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:133"/>
 <a href="polynom1.html#K17" target="_self">BagOrder</a> <font color="Maroon">n</font> <a href="relat_2.html#R6">is_connected_in</a>  <a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="133_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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_2.html#D6">RELAT_2:def 6</a><br/>

<b>assume </b><a NAME="E1:133_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> &amp; 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="polynom1.html#K17" target="_self">BagOrder</a> <font color="Maroon">n</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">p</font> = <font color="Maroon">x</font>, <font color="Maroon">q</font> = <font color="Maroon">y</font> as   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> <b>by </b><i><a class="ref" href="polynom1.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E3:133_1"/>
not <font color="Maroon">p</font> <a href="polynom1.html#R2" target="_self">&lt;='</a> <font color="Maroon">q</font>
 
<b>by </b><i>, <a class="ref" href="polynom1.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E4:133_1"/><b>then </b>
<font color="Maroon">q</font> <a href="polynom1.html#R2" target="_self">&lt;='</a> <font color="Maroon">p</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:133_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="polynom1.html#K17" target="_self">BagOrder</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T8" target="_self">Th8</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:133"/>
 <a href="polynom1.html#K17" target="_self">BagOrder</a> <font color="Maroon">n</font> <a href="orders_1.html#R3">linearly_orders</a>  <a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T9" target="_self">Th9</a>, , <a class="ref" href="polynom1.html#T10" target="_self">Th10</a>, <a class="ref" href="orders_1.html#D8">ORDERS_1:def 8</a></i>;<br/>


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