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<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">L</font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="group_1.html#V2">unital</a> <a href="vectsp_1.html#V7">distributive</a> non <a href="realset2.html#V3">trivial</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">p</font> be   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>;<br/>





<b>set </b><font color="Maroon">R</font> =  <a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font>;<br/>
<a NAME="E1:41"/>
(  <a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font> is <a href="relat_2.html#V1">reflexive</a> &amp;  <a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font> is <a href="relat_2.html#V8">transitive</a> &amp;  <a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font> is <a href="relat_2.html#V4">antisymmetric</a> &amp;  <a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font> is <a href="relat_2.html#V6">connected</a> )
 
<b>by </b><i><a class="ref" href="orders_1.html#D5">ORDERS_1:def 5</a></i>;<br/>
<a NAME="E2:41"/><b>then </b><i><font color="Green">E32</font></i>: 
(  <a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font> <a href="relat_2.html#R1">is_reflexive_in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span> &amp;  <a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font> <a href="relat_2.html#R4">is_antisymmetric_in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span> &amp;  <a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font> <a href="relat_2.html#R8">is_transitive_in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span> &amp;  <a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font> <a href="relat_2.html#R6">is_connected_in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span> )
 
<b>by </b><i><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#D14">RELAT_2:def 14</a>, <a class="ref" href="relat_2.html#D16">RELAT_2:def 16</a></i>;<br/>
<a NAME="E3:41"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:41_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</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#D14">POLYNOM1:def 14</a></i>;<br/>
<a NAME="E3:41_1"/>
 <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font> <a href="polynom1.html#R2">&lt;='</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#T64">POLYNOM1:64</a></i>;<br/>
<a NAME="E4:41_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font></span>)</span>,<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">BagOrder</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#D16">POLYNOM1:def 16</a></i>;<br/>
<a NAME="E5:41_1"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#D5">RELAT_1:def 5</a></i>;<br/>
<a NAME="E6:41_1"/>
 <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#D6">RELAT_1:def 6</a></i>;<br/>
<a NAME="E7:41_1"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<b>hence </b><a NAME="E8:41_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E4:41"/><b>then </b><i><font color="Green">E35</font></i>: 
 <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E5:41"/><b>then </b><i><font color="Green">E36</font></i>: 
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E6:41"/><i><font color="Green">E37</font></i>: 
(  <a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font> <a href="relat_2.html#R1">is_reflexive_in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font> &amp;  <a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font> <a href="relat_2.html#R4">is_antisymmetric_in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font> &amp;  <a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font> <a href="relat_2.html#R8">is_transitive_in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E7:41"/>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K3">field</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K3">field</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E8:41"/><b>then </b>
 <a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K3">field</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K3">field</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">R'</font> =  <a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font> as   <a href="relset_1.html#NM3">Relation</a> of  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i><a class="ref" href="relset_1.html#D1">RELSET_1:def 1</a></i>;<br/>
<a NAME="E10:41"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">R'</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="orders_1.html#T98">ORDERS_1:98</a></i>;<br/>
<a NAME="E11:41"/><b>then </b>
<font color="Maroon">R'</font> is <a href="partfun1.html#V1">total</a>
 
<b>by </b><i><a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a></i>;<br/>
<a NAME="E12:41"/><b>then </b>
<font color="Maroon">R'</font> <a href="relat_2.html#R6">is_connected_in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E13:41"/>
 <a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font> <a href="orders_1.html#R3">linearly_orders</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="txt" href="polynom2.html#E2:41"><i><font color="Green">E32</font></i></a>, <a class="ref" href="orders_1.html#D8">ORDERS_1:def 8</a></i>;<br/>


</div>
