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<b>defpred </b><font color="Maroon">S<sub>1</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> ex <font color="Olive" title="b1">b1</font>, <font color="Olive" title="b2">b2</font> being   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b1</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b2</font> &amp; <font color="Olive" title="b1">b1</font> <a href="polynom1.html#R2" title="POLYNOM1:pred.2">&lt;='</a> <font color="Olive" title="b2">b2</font> );<br/>
<b>consider </b><font color="Maroon" title="c2">BO</font> being    <a href="relset_1.html#M2" title="RELSET_1:mode.2">Relation</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font>, <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E2:128_1_1"/><i><font color="Green" title="E66">A1</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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">x</font>,<font color="Olive" title="b2">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="c2">BO</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] ) )
 <b>from </b><i><a class="ref" href="relset_1.html#S1" title="RELSET_1:sch.1">RELSET_1:sch 1</a>();<br/></i>
<a NAME="E3:128_1_1"/><i><font color="Green" title="E67">A2</font></i>: 
<font color="Maroon" title="c2">BO</font> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a>  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="128_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">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="E1:128_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font>
 ;<br/>

<b>hence </b><a NAME="E2:128_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">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="c2">BO</font>
 <b>by </b><i><a class="txt" href="polynom1.html#E2:128_1_1"><i><font color="Green" title="E66">A1</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:128_1_1"/><i><font color="Green" title="E68">A3</font></i>: 
<font color="Maroon" title="c2">BO</font> <a href="relat_2.html#R4" title="RELAT_2:pred.4">is_antisymmetric_in</a>  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="128_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">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_2.html#D4" title="RELAT_2:def.4">RELAT_2:def 4</a><br/>

<b>assume </b><a NAME="E1:128_1_1_2"/><i><font color="Green" title="E69">A4</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">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="c2">BO</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c3">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="c2">BO</font> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">b1</font>, <font color="Maroon" title="c6">b2</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E3:128_1_1_2"/><i><font color="Green" title="E70">A5</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b1</font> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b2</font> &amp; <font color="Maroon" title="c5">b1</font> <a href="polynom1.html#R2" title="POLYNOM1:pred.2">&lt;='</a> <font color="Maroon" title="c6">b2</font> )
 <b>by </b><i><a class="txt" href="polynom1.html#E2:128_1_1"><i><font color="Green" title="E66">A1</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">b1'</font>, <font color="Maroon" title="c8">b2'</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E5:128_1_1_2"/><i><font color="Green" title="E71">A6</font></i>: 
( <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b1'</font> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b2'</font> &amp; <font color="Maroon" title="c7">b1'</font> <a href="polynom1.html#R2" title="POLYNOM1:pred.2">&lt;='</a> <font color="Maroon" title="c8">b2'</font> )
 <b>by </b><i><a class="txt" href="polynom1.html#E2:128_1_1"><i><font color="Green" title="E66">A1</font></i></a>, <a class="txt" href="polynom1.html#E1:128_1_1_2"><i><font color="Green" title="E69">A4</font></i></a></i>;<br/>
<a NAME="E6:128_1_1_2"/>
( ( <font color="Maroon" title="c5">b1</font> <a href="polynom1.html#R1" title="POLYNOM1:pred.1">&lt;</a> <font color="Maroon" title="c6">b2</font> or <font color="Maroon" title="c5">b1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c6">b2</font> ) &amp; ( <font color="Maroon" title="c7">b1'</font> <a href="polynom1.html#R1" title="POLYNOM1:pred.1">&lt;</a> <font color="Maroon" title="c8">b2'</font> or <font color="Maroon" title="c7">b1'</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c8">b2'</font> ) )
 
<b>by </b><i><a class="txt" href="polynom1.html#E3:128_1_1_2"><i><font color="Green" title="E70">A5</font></i></a>, <a class="txt" href="polynom1.html#E5:128_1_1_2"><i><font color="Green" title="E71">A6</font></i></a>, <a class="ref" href="polynom1.html#D12" target="_self" title="POLYNOM1:def.12">Def12</a></i>;<br/>
<b>hence </b><a NAME="E7:128_1_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font>
 <b>by </b><i><a class="txt" href="polynom1.html#E3:128_1_1_2"><i><font color="Green" title="E70">A5</font></i></a>, <a class="txt" href="polynom1.html#E5:128_1_1_2"><i><font color="Green" title="E71">A6</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:128_1_1"/><i><font color="Green" title="E69">A7</font></i>: 
<font color="Maroon" title="c2">BO</font> <a href="relat_2.html#R8" title="RELAT_2:pred.8">is_transitive_in</a>  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="128_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font>, <font color="Maroon" title="c5">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#D8" title="RELAT_2:def.8">RELAT_2:def 8</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:128_1_1_3"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font> )
 <b>and </b><br/><a NAME="E2:128_1_1_3"/><i><font color="Green" title="E70">A8</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">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="c2">BO</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">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="c2">BO</font> )
 ;<br/>

<b>consider </b><font color="Maroon" title="c6">b1</font>, <font color="Maroon" title="c7">b2</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E4:128_1_1_3"/><i><font color="Green" title="E71">A9</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b1</font> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b2</font> &amp; <font color="Maroon" title="c6">b1</font> <a href="polynom1.html#R2" title="POLYNOM1:pred.2">&lt;='</a> <font color="Maroon" title="c7">b2</font> )
 <b>by </b><i><a class="txt" href="polynom1.html#E2:128_1_1"><i><font color="Green" title="E66">A1</font></i></a>, <a class="txt" href="polynom1.html#E2:128_1_1_3"><i><font color="Green" title="E70">A8</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">b1'</font>, <font color="Maroon" title="c9">b2'</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E6:128_1_1_3"/><i><font color="Green" title="E72">A10</font></i>: 
( <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b1'</font> &amp; <font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b2'</font> &amp; <font color="Maroon" title="c8">b1'</font> <a href="polynom1.html#R2" title="POLYNOM1:pred.2">&lt;='</a> <font color="Maroon" title="c9">b2'</font> )
 <b>by </b><i><a class="txt" href="polynom1.html#E2:128_1_1"><i><font color="Green" title="E66">A1</font></i></a>, <a class="txt" href="polynom1.html#E2:128_1_1_3"><i><font color="Green" title="E70">A8</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">B1</font> = <font color="Maroon" title="c6">b1</font>, <font color="Maroon" title="c11">B2'</font> = <font color="Maroon" title="c9">b2'</font> as   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font> ;<br/>
<a NAME="E8:128_1_1_3"/>
<font color="Maroon" title="c10">B1</font> <a href="polynom1.html#R2" title="POLYNOM1:pred.2">&lt;='</a> <font color="Maroon" title="c11">B2'</font>
 
<b>by </b><i><a class="txt" href="polynom1.html#E4:128_1_1_3"><i><font color="Green" title="E71">A9</font></i></a>, <a class="txt" href="polynom1.html#E6:128_1_1_3"><i><font color="Green" title="E72">A10</font></i></a>, <a class="ref" href="polynom1.html#T46" target="_self" title="POLYNOM1:th.46">Th46</a></i>;<br/>
<b>hence </b><a NAME="E9:128_1_1_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">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="c2">BO</font>
 <b>by </b><i><a class="txt" href="polynom1.html#E2:128_1_1"><i><font color="Green" title="E66">A1</font></i></a>, <a class="txt" href="polynom1.html#E4:128_1_1_3"><i><font color="Green" title="E71">A9</font></i></a>, <a class="txt" href="polynom1.html#E6:128_1_1_3"><i><font color="Green" title="E72">A10</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:128_1_1"/>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">BO</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c2">BO</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font> )
 
<b>by </b><i><a class="txt" href="polynom1.html#E3:128_1_1"><i><font color="Green" title="E67">A2</font></i></a>, <a class="ref" href="orders_1.html#T98" title="ORDERS_1:th.98">ORDERS_1:98</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c3">BO</font> = <font color="Maroon" title="c2">BO</font> as   <a href="orders_1.html#NM2" title="ORDERS_1:NM.2">Order</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font> <b>by </b><i><a class="txt" href="polynom1.html#E3:128_1_1"><i><font color="Green" title="E67">A2</font></i></a>, <a class="txt" href="polynom1.html#E4:128_1_1"><i><font color="Green" title="E68">A3</font></i></a>, <a class="txt" href="polynom1.html#E5:128_1_1"><i><font color="Green" title="E69">A7</font></i></a>, <a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a>, <a class="ref" href="relat_2.html#D9" title="RELAT_2:def.9">RELAT_2:def 9</a>, <a class="ref" href="relat_2.html#D12" title="RELAT_2:def.12">RELAT_2:def 12</a>, <a class="ref" href="relat_2.html#D16" title="RELAT_2:def.16">RELAT_2:def 16</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c3">BO</font>
;<br/>

<b>let </b><font color="Maroon" title="c4">p</font>, <font color="Maroon" title="c5">q</font> be   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>;<br/>

<a NAME="E8:128_1_1"/><i><font color="Green" title="E70">A11</font></i>: 
( <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font> )
 
<b>by </b><i><a class="ref" href="polynom1.html#D14" target="_self" title="POLYNOM1:def.14">Def14</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:128_1_1_4"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</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="c3">BO</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c6">b1</font>, <font color="Maroon" title="c7">b2</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E3:128_1_1_4"/><i><font color="Green" title="E71">A12</font></i>: 
( <font color="Maroon" title="c4">p</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c6">b1</font> &amp; <font color="Maroon" title="c5">q</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c7">b2</font> &amp; <font color="Maroon" title="c6">b1</font> <a href="polynom1.html#R2" title="POLYNOM1:pred.2">&lt;='</a> <font color="Maroon" title="c7">b2</font> )
 <b>by </b><i><a class="txt" href="polynom1.html#E2:128_1_1"><i><font color="Green" title="E66">A1</font></i></a></i>;<br/><b>thus </b><a NAME="E4:128_1_1_4"/>
<font color="Maroon" title="c4">p</font> <a href="polynom1.html#R2" title="POLYNOM1:pred.2">&lt;='</a> <font color="Maroon" title="c5">q</font>
 <b>by </b><i><a class="txt" href="polynom1.html#E3:128_1_1_4"><i><font color="Green" title="E71">A12</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E10:128_1_1"/>
( <font color="Maroon" title="c4">p</font> <a href="polynom1.html#R2" title="POLYNOM1:pred.2">&lt;='</a> <font color="Maroon" title="c5">q</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</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="c3">BO</font> )
 <b>by </b><i><a class="txt" href="polynom1.html#E2:128_1_1"><i><font color="Green" title="E66">A1</font></i></a>, <a class="txt" href="polynom1.html#E8:128_1_1"><i><font color="Green" title="E70">A11</font></i></a></i>;<br/>


</div>
