<?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">T</font> be  <a href="relat_2.html#V6">connected</a> <a href="bagorder.html#V2">admissible</a> <a href="bagorder.html#NM1">TermOrder</a> of <font color="Maroon">n</font>;<br/><b>let </b><font color="Maroon">L</font> be  <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> non <a href="realset2.html#V3">trivial</a>  <a href="rlvect_1.html#L1">LoopStr</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>assume </b><a NAME="E1:51"/><i><font color="Green">E23</font></i>: 
 <a href="termord.html#K3">HT</a> <font color="Maroon">p</font>,<font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font>
 ;<br/>

<div><i><font color="Green">E24</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">b</font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>;<br/><b>assume </b><a NAME="E1:51_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E2:51_1"/>
<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">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="bagorder.html#D7">BAGORDER:def 7</a></i>;<br/><b>hence </b><a NAME="E3:51_1"/>
<font color="Maroon">p</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 <b>by </b><i>, , <a class="txt" href="polyred.html#E4"><i><font color="Green">Lemma40</font></i></a></i>;<br/></div><b>end;</b></div>
<b>set </b><font color="Maroon">red</font> =  <a href="termord.html#K6">Red</a> <font color="Maroon">p</font>,<font color="Maroon">T</font>;<br/>
<b>set </b><font color="Maroon">e</font> =  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font>;<br/>
<i><font color="Green">E27</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="termord.html#K6">Red</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span> = 
 <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:51_3"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="termord.html#K6">Red</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">b'</font> = <font color="Maroon">b</font> as    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font> ;<br/><i><font color="Green">E38</font></i>: <span class="p1">(<span class="default"><a href="termord.html#K6">Red</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b'</font> = 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K25">-</a> <span class="p2">(<span class="default"><a href="termord.html#K5">HM</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b'</font>
<b>by </b><i><a class="ref" href="termord.html#D9">TERMORD:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K22">+</a> <span class="p2">(<span class="default"><a href="polynom1.html#K24">-</a> <span class="p3">(<span class="default"><a href="termord.html#K5">HM</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b'</font>
<b>by </b><i><a class="ref" href="polynom1.html#D23">POLYNOM1:def 23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b'</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K24">-</a> <span class="p3">(<span class="default"><a href="termord.html#K5">HM</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b'</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D21">POLYNOM1:def 21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b'</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="termord.html#K5">HM</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b'</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D22">POLYNOM1:def 22</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_3_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:51_3_2_1"/>
( <font color="Maroon">b'</font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font> or <font color="Maroon">b'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_2_1_1"><b>case </b></a><a NAME="E1:51_3_2_1_1"/>
<font color="Maroon">b'</font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><a href="termord.html#K6">Red</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b'</font> = 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15">.</a> <span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15">.</a> <span class="p3">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="termord.html#T18">TERMORD:18</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T16">RLVECT_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b'</font>
<b>by </b><i><a class="ref" href="polynom1.html#T81">POLYNOM1:81</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_2_1_2"><b>case </b></a><a NAME="E1:51_3_2_1_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">b'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><a href="termord.html#K6">Red</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b'</font> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="termord.html#K5">HM</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b'</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="termord.html#T19">TERMORD:19</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T16">RLVECT_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b'</font>
<b>by </b><i><a class="ref" href="polynom1.html#T81">POLYNOM1:81</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:51_3"/>
<span class="p1">(<span class="default"><a href="termord.html#K6">Red</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:51"/>
 <a href="termord.html#K6">Red</a> <font color="Maroon">p</font>,<font color="Maroon">T</font> <a href="funct_2.html#R2">=</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div>
