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

<b>let </b><font color="Maroon" title="c1">n</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/><b>let </b><font color="Maroon" title="c2">T</font> be  <a href="relat_2.html#V6" title="RELAT_2:attr.6">connected</a> <a href="bagorder.html#V2" title="BAGORDER:attr.2">admissible</a> <a href="bagorder.html#NM1" title="BAGORDER:NM.1">TermOrder</a> of <font color="Maroon" title="c1">n</font>;<br/><b>let </b><font color="Maroon" title="c3">L</font> be  non <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">add-associative</a> <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">right_zeroed</a> <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">right_complementable</a>  <a href="rlvect_1.html#L1" title="RLVECT_1:struct.1">LoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c4">p</font>, <font color="Maroon" title="c5">q</font> be   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>;<br/>







<b>assume </b><a NAME="E1:53"/><i><font color="Green" title="E44">A1</font></i>: 
<font color="Maroon" title="c5">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K26" title="POLYNOM1:func.26">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c6">R</font> =  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">T</font> #);<br/>
<b>assume </b><a NAME="E2:53"/><i><font color="Green" title="E45">A2</font></i>: 
(  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c2">T</font> &amp;  <a href="termord.html#K6" title="TERMORD:func.6">Red</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font> <a href="polyred.html#R1" title="POLYRED:pred.1">&lt;=</a>  <a href="termord.html#K6" title="TERMORD:func.6">Red</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c2">T</font> )
 ;<br/>

<a NAME="E3:53"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p3">(<span class="default"><a href="termord.html#K6" title="TERMORD:func.6">Red</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p3">(<span class="default"><a href="termord.html#K6" title="TERMORD:func.6">Red</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">T</font> #)
 
<b>by </b><i><a class="ref" href="polyred.html#D2" target="_self" title="POLYRED:def.2">Def2</a></i>;<br/>
<a NAME="E4:53"/><b>then </b><i><font color="Green" title="E46">A3</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p3">(<span class="default"><a href="termord.html#K6" title="TERMORD:func.6">Red</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p3">(<span class="default"><a href="termord.html#K6" title="TERMORD:func.6">Red</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="bagorder.html#K12" title="BAGORDER:func.12">FinOrd-Approx</a> <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">T</font> #)</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="bagorder.html#D17" title="BAGORDER:def.17">BAGORDER:def 17</a></i>;<br/>
<b>set </b><font color="Maroon" title="c7">rp</font> =  <a href="termord.html#K6" title="TERMORD:func.6">Red</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>;<br/>
<b>set </b><font color="Maroon" title="c8">rq</font> =  <a href="termord.html#K6" title="TERMORD:func.6">Red</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c2">T</font>;<br/>
<b>set </b><font color="Maroon" title="c9">x</font> =  <a href="polyred.html#K2" title="POLYRED:func.2">Support</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>;<br/>
<b>set </b><font color="Maroon" title="c10">y</font> =  <a href="polyred.html#K2" title="POLYRED:func.2">Support</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c2">T</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_1_1"/>
( <font color="Maroon" title="c4">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="polynom1.html#K26" title="POLYNOM1:func.26">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> or <font color="Maroon" title="c4">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K26" title="POLYNOM1:func.26">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_1"><b>case </b></a><a NAME="E1:53_1_1_1"/>
<font color="Maroon" title="c4">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="polynom1.html#K26" title="POLYNOM1:func.26">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:53_1_1_1"/>
<font color="Maroon" title="c4">p</font> <a href="polyred.html#R1" title="POLYRED:pred.1">&lt;=</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="txt" href="polyred.html#E41"><i><font color="Green" title="E35">Lm12</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_2"><b>case </b></a><a NAME="E1:53_1_1_2"/><i><font color="Green" title="E47">A4</font></i>: 
<font color="Maroon" title="c4">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K26" title="POLYNOM1:func.26">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>
 ;<br/><div class="add"><a NAME="E2:53_1_1_2"/><b>then </b><i><font color="Green" title="E48">A5</font></i>: 
<font color="Maroon" title="c4">p</font> is <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a>
 <b>by </b><i><a class="ref" href="polynom7.html#D2" title="POLYNOM7:def.2">POLYNOM7:def 2</a></i>;<br/><a NAME="E3:53_1_1_2"/><i><font color="Green" title="E49">A6</font></i>: 
<font color="Maroon" title="c5">q</font> is <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a>
 <b>by </b><i><a class="txt" href="polyred.html#E1:53"><i><font color="Green" title="E44">A1</font></i></a>, <a class="ref" href="polynom7.html#D2" title="POLYNOM7:def.2">POLYNOM7:def 2</a></i>;<br/><a NAME="E4:53_1_1_2"/><i><font color="Green" title="E50">A7</font></i>: 
(  <a href="polyred.html#K2" title="POLYRED:func.2">Support</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="polyred.html#K2" title="POLYRED:func.2">Support</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c2">T</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <b>by </b><i><a class="txt" href="polyred.html#E1:53"><i><font color="Green" title="E44">A1</font></i></a>, <a class="txt" href="polyred.html#E1:53_1_1_2"><i><font color="Green" title="E47">A4</font></i></a>, <a class="ref" href="polynom7.html#T1" title="POLYNOM7:th.1">POLYNOM7:1</a></i>;<br/><i><font color="Green" title="E51">A8</font></i>:  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <span class="p1">(<span class="default"><a href="polyred.html#K2" title="POLYRED:func.2">Support</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font></span>)</span> = 
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c2">T</font>
<b>by </b><i><a class="txt" href="polyred.html#E2:53"><i><font color="Green" title="E45">A2</font></i></a>, <a class="txt" href="polyred.html#E2:53_1_1_2"><i><font color="Green" title="E48">A5</font></i></a>, <a class="ref" href="polyred.html#T24" target="_self" title="POLYRED:th.24">Th24</a></i>
<br/>.= 
 <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <span class="p1">(<span class="default"><a href="polyred.html#K2" title="POLYRED:func.2">Support</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c2">T</font></span>)</span>
<b>by </b><i><a class="txt" href="polyred.html#E3:53_1_1_2"><i><font color="Green" title="E49">A6</font></i></a>, <a class="ref" href="polyred.html#T24" target="_self" title="POLYRED:th.24">Th24</a></i>
;<br/><i><font color="Green" title="E52">A9</font></i>:  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="termord.html#K6" title="TERMORD:func.6">Red</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="ref" href="termord.html#T36" title="TERMORD:th.36">TERMORD:36</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polyred.html#K2" title="POLYRED:func.2">Support</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <span class="p3">(<span class="default"><a href="polyred.html#K2" title="POLYRED:func.2">Support</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="txt" href="polyred.html#E2:53_1_1_2"><i><font color="Green" title="E48">A5</font></i></a>, <a class="ref" href="polyred.html#T24" target="_self" title="POLYRED:th.24">Th24</a></i>
;<br/> <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="termord.html#K6" title="TERMORD:func.6">Red</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c2">T</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c2">T</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="ref" href="termord.html#T36" title="TERMORD:th.36">TERMORD:36</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polyred.html#K2" title="POLYRED:func.2">Support</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c2">T</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <span class="p3">(<span class="default"><a href="polyred.html#K2" title="POLYRED:func.2">Support</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="txt" href="polyred.html#E3:53_1_1_2"><i><font color="Green" title="E49">A6</font></i></a>, <a class="ref" href="polyred.html#T24" target="_self" title="POLYRED:th.24">Th24</a></i>
;<br/><a NAME="E8:53_1_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="polyred.html#K2" title="POLYRED:func.2">Support</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font></span>)</span>,<span class="p2">(<span class="default"><a href="polyred.html#K2" title="POLYRED:func.2">Support</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c2">T</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="bagorder.html#K12" title="BAGORDER:func.12">FinOrd-Approx</a> <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">T</font> #)</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="polyred.html#E4:53"><i><font color="Green" title="E46">A3</font></i></a>, <a class="txt" href="polyred.html#E4:53_1_1_2"><i><font color="Green" title="E50">A7</font></i></a>, <a class="txt" href="polyred.html#E5:53_1_1_2"><i><font color="Green" title="E51">A8</font></i></a>, <a class="txt" href="polyred.html#E6:53_1_1_2"><i><font color="Green" title="E52">A9</font></i></a>, <a class="ref" href="bagorder.html#T36" title="BAGORDER:th.36">BAGORDER:36</a></i>;<br/><a NAME="E9:53_1_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="polyred.html#K2" title="POLYRED:func.2">Support</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font></span>)</span>,<span class="p2">(<span class="default"><a href="polyred.html#K2" title="POLYRED:func.2">Support</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c2">T</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">T</font> #)
 <b>by </b><i><a class="ref" href="bagorder.html#D17" title="BAGORDER:def.17">BAGORDER:def 17</a></i>;<br/><b>hence </b><a NAME="E10:53_1_1_2"/>
<font color="Maroon" title="c4">p</font> <a href="polyred.html#R1" title="POLYRED:pred.1">&lt;=</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="ref" href="polyred.html#D2" target="_self" title="POLYRED:def.2">Def2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:53"/>
<font color="Maroon" title="c4">p</font> <a href="polyred.html#R1" title="POLYRED:pred.1">&lt;=</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c2">T</font>
 ;<br/>


</div>
