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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">b1</font>, <font color="Olive">b2</font> being   <a href="polynom1.html#M1" target="_self">Element</a> of  <a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b1</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b2</font> &amp; <font color="Olive">b1</font> <a href="polynom1.html#R2" target="_self">&lt;='</a> <font color="Olive">b2</font> );<br/>
<b>consider </b><font color="Maroon">BO</font> being    <a href="relset_1.html#M2">Relation</a> of  <a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font>, <a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E2:129_1_1"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">BO</font> iff ( <font color="Olive">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="Olive">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">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] ) )
 <b>from </b><i><a class="ref" href="relset_1.html#S1">RELSET_1:sch 1</a>();<br/></i>
<a NAME="E3:129_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">BO</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="129_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:129_1_1_1"/>
<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>
 ;<br/>

<b>hence </b><a NAME="E2:129_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">BO</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T8" target="_self">Th8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:129_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">BO</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="129_1_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:129_1_1_2"/><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; <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">BO</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">BO</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">b1</font> being    <a href="polynom1.html#M1" target="_self">Element</a> of  <a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font>, <font color="Maroon">b2</font> being    <a href="polynom1.html#M1" target="_self">Element</a> of  <a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E3:129_1_1_2"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> &amp; <font color="Maroon">b1</font> <a href="polynom1.html#R2" target="_self">&lt;='</a> <font color="Maroon">b2</font> )
 <b>by </b><i><a class="ref" href="polynom1.html#T8" target="_self">Th8</a></i>;<br/>
<b>consider </b><font color="Maroon">b1'</font> being    <a href="polynom1.html#M1" target="_self">Element</a> of  <a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font>, <font color="Maroon">b2'</font> being    <a href="polynom1.html#M1" target="_self">Element</a> of  <a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E5:129_1_1_2"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1'</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2'</font> &amp; <font color="Maroon">b1'</font> <a href="polynom1.html#R2" target="_self">&lt;='</a> <font color="Maroon">b2'</font> )
 <b>by </b><i><a class="ref" href="polynom1.html#T8" target="_self">Th8</a>, </i>;<br/>
<a NAME="E6:129_1_1_2"/>
( ( <font color="Maroon">b1</font> <a href="polynom1.html#R1" target="_self">&lt;</a> <font color="Maroon">b2</font> or <font color="Maroon">b1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b2</font> ) &amp; ( <font color="Maroon">b1'</font> <a href="polynom1.html#R1" target="_self">&lt;</a> <font color="Maroon">b2'</font> or <font color="Maroon">b1'</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b2'</font> ) )
 
<b>by </b><i><a class="txt" href="polynom1.html#E8:129_1_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="polynom1.html#T9" target="_self">Th9</a>, </i>;<br/>
<b>hence </b><a NAME="E7:129_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="txt" href="polynom1.html#E8:129_1_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="polynom1.html#T9" target="_self">Th9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:129_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">BO</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="129_1_1_3"><b>proof </b></a><div class="add">

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

<b>assume </b><b>that </b><br/><a NAME="E1:129_1_1_3"/>
( <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">z</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font> )
 <b>and </b><br/><a NAME="E2:129_1_1_3"/><i><font color="Green">E69</font></i>: 
( <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">BO</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">BO</font> )
 ;<br/>

<b>consider </b><font color="Maroon">b1</font> being    <a href="polynom1.html#M1" target="_self">Element</a> of  <a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font>, <font color="Maroon">b2</font> being    <a href="polynom1.html#M1" target="_self">Element</a> of  <a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E4:129_1_1_3"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> &amp; <font color="Maroon">b1</font> <a href="polynom1.html#R2" target="_self">&lt;='</a> <font color="Maroon">b2</font> )
 <b>by </b><i><a class="ref" href="polynom1.html#T8" target="_self">Th8</a>, <a class="txt" href="polynom1.html#E8:129_1_1"><i><font color="Green">E72</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">b1'</font> being    <a href="polynom1.html#M1" target="_self">Element</a> of  <a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font>, <font color="Maroon">b2'</font> being    <a href="polynom1.html#M1" target="_self">Element</a> of  <a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E6:129_1_1_3"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1'</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2'</font> &amp; <font color="Maroon">b1'</font> <a href="polynom1.html#R2" target="_self">&lt;='</a> <font color="Maroon">b2'</font> )
 <b>by </b><i><a class="ref" href="polynom1.html#T8" target="_self">Th8</a>, <a class="txt" href="polynom1.html#E8:129_1_1"><i><font color="Green">E72</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">B1</font> = <font color="Maroon">b1</font>, <font color="Maroon">B2'</font> = <font color="Maroon">b2'</font> as   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> ;<br/>
<a NAME="E8:129_1_1_3"/>
<font color="Maroon">B1</font> <a href="polynom1.html#R2" target="_self">&lt;='</a> <font color="Maroon">B2'</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#T9" target="_self">Th9</a>, , <a class="ref" href="polynom1.html#T4" target="_self">Th4</a></i>;<br/>
<b>hence </b><a NAME="E9:129_1_1_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">BO</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T8" target="_self">Th8</a>, <a class="ref" href="polynom1.html#T9" target="_self">Th9</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:129_1_1"/>
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">BO</font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font> &amp;  <a href="relat_1.html#K3">field</a> <font color="Maroon">BO</font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font> )
 
<b>by </b><i>, <a class="ref" href="orders_1.html#T98">ORDERS_1:98</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">BO</font> = <font color="Maroon">BO</font> as   <a href="orders_1.html#NM2">Order</a> of  <a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font> <b>by </b><i>, , , <a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a>, <a class="ref" href="relat_2.html#D9">RELAT_2:def 9</a>, <a class="ref" href="relat_2.html#D12">RELAT_2:def 12</a>, <a class="ref" href="relat_2.html#D16">RELAT_2:def 16</a></i>;<br/>
<b>take </b>
<font color="Maroon">BO</font>
;<br/>

<b>let </b><font color="Maroon">p</font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>, <font color="Maroon">q</font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>;<br/>

<a NAME="E8:129_1_1"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">p</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">q</font> <a href="hidden.html#R2">in</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#T6" target="_self">Th6</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:129_1_1_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">BO</font>
 ;<br/><b>then consider </b><font color="Maroon">b1</font> being    <a href="polynom1.html#M1" target="_self">Element</a> of  <a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font>, <font color="Maroon">b2</font> being    <a href="polynom1.html#M1" target="_self">Element</a> of  <a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E3:129_1_1_4"/><i><font color="Green">E108</font></i>: 
( <font color="Maroon">p</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">q</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b2</font> &amp; <font color="Maroon">b1</font> <a href="polynom1.html#R2" target="_self">&lt;='</a> <font color="Maroon">b2</font> )
 <b>by </b><i><a class="ref" href="polynom1.html#T8" target="_self">Th8</a></i>;<br/><b>thus </b><a NAME="E4:129_1_1_4"/>
<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#T9" target="_self">Th9</a></i>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E10:129_1_1"/>
( <font color="Maroon">p</font> <a href="polynom1.html#R2" target="_self">&lt;='</a> <font color="Maroon">q</font> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">BO</font> )
 <b>by </b><i><a class="ref" href="polynom1.html#T8" target="_self">Th8</a>, <a class="txt" href="polynom1.html#E8:129_1_1"><i><font color="Green">E72</font></i></a></i>;<br/>


</div>
