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

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</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  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V3">Abelian</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="group_1.html#V4">associative</a> <a href="group_1.html#V7">commutative</a> <a href="vectsp_1.html#V7">distributive</a> <a href="vectsp_1.html#V9">Field-like</a> non <a href="vectsp_1.html#V10">degenerated</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span>;<br/>







<b>assume </b><a NAME="E1:25"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">f</font> being <a href="polynom7.html#V1">non-zero</a> <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>  st <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="ideal_1.html#K7">-Ideal</a>  holds <br/><font color="Olive">f</font> <a href="polyred.html#R7">is_reducible_wrt</a> <font color="Maroon">P</font>,<font color="Maroon">T</font>
 ;<br/>

<b>thus </b><a NAME="E2:25"/>
for <font color="Olive">f</font> being <a href="polynom7.html#V1">non-zero</a> <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>  st <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="ideal_1.html#K7">-Ideal</a>  holds <br/><font color="Olive">f</font> <a href="polyred.html#R10">is_top_reducible_wrt</a> <font color="Maroon">P</font>,<font color="Maroon">T</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be  <a href="polynom7.html#V1">non-zero</a> <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>;<br/>

<b>assume </b><a NAME="E1:25_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="ideal_1.html#K7">-Ideal</a> 
 ;<br/>

<b>set </b><font color="Maroon">H</font> = <span class="p1">{<span class="default"> <font color="Olive">g</font> where <font color="Olive">g</font> is  <a href="polynom7.html#V1">non-zero</a> <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> : ( <font color="Olive">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="ideal_1.html#K7">-Ideal</a>  &amp; not <font color="Olive">g</font> <a href="polyred.html#R10">is_top_reducible_wrt</a> <font color="Maroon">P</font>,<font color="Maroon">T</font> ) </span>}</span> ;<br/>
<b>assume </b><a NAME="E2:25_1"/>
not <font color="Maroon">f</font> <a href="polyred.html#R10">is_top_reducible_wrt</a> <font color="Maroon">P</font>,<font color="Maroon">T</font>
 ;<br/>

<a NAME="E3:25_1"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">g</font> where <font color="Olive">g</font> is  <a href="polynom7.html#V1">non-zero</a> <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> : ( <font color="Olive">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="ideal_1.html#K7">-Ideal</a>  &amp; not <font color="Olive">g</font> <a href="polyred.html#R10">is_top_reducible_wrt</a> <font color="Maroon">P</font>,<font color="Maroon">T</font> ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:25_1_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">g</font> where <font color="Olive">g</font> is  <a href="polynom7.html#V1">non-zero</a> <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> : ( <font color="Olive">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="ideal_1.html#K7">-Ideal</a>  &amp; not <font color="Olive">g</font> <a href="polyred.html#R10">is_top_reducible_wrt</a> <font color="Maroon">P</font>,<font color="Maroon">T</font> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">g'</font> being  <a href="polynom7.html#V1">non-zero</a> <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:25_1_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g'</font> &amp; <font color="Maroon">g'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="ideal_1.html#K7">-Ideal</a>  &amp; not <font color="Maroon">g'</font> <a href="polyred.html#R10">is_top_reducible_wrt</a> <font color="Maroon">P</font>,<font color="Maroon">T</font> )
 ;<br/><b>thus </b><a NAME="E4:25_1_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">H</font> = <span class="p1">{<span class="default"> <font color="Olive">g</font> where <font color="Olive">g</font> is  <a href="polynom7.html#V1">non-zero</a> <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> : ( <font color="Olive">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="ideal_1.html#K7">-Ideal</a>  &amp; not <font color="Olive">g</font> <a href="polyred.html#R10">is_top_reducible_wrt</a> <font color="Maroon">P</font>,<font color="Maroon">T</font> ) </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span> <b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">p</font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E7:25_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> &amp; ( for <font color="Olive">q</font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>  st <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> holds <br/><font color="Maroon">p</font> <a href="polyred.html#R1">&lt;=</a> <font color="Olive">q</font>,<font color="Maroon">T</font> ) )
 <b>by </b><i><a class="ref" href="polyred.html#T31">POLYRED:31</a></i>;<br/>
<b>consider </b><font color="Maroon">p'</font> being  <a href="polynom7.html#V1">non-zero</a> <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E9:25_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">p'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; <font color="Maroon">p'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="ideal_1.html#K7">-Ideal</a>  &amp; not <font color="Maroon">p'</font> <a href="polyred.html#R10">is_top_reducible_wrt</a> <font color="Maroon">P</font>,<font color="Maroon">T</font> )
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p</font> as  <a href="polynom7.html#V1">non-zero</a> <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> <b>by </b><i/>;<br/>
<a NAME="E11:25_1"/>
<font color="Maroon">p</font> <a href="polyred.html#R7">is_reducible_wrt</a> <font color="Maroon">P</font>,<font color="Maroon">T</font>
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">q</font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E13:25_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">p</font> <a href="polyred.html#R5">reduces_to</a> <font color="Maroon">q</font>,<font color="Maroon">P</font>,<font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="polyred.html#D9">POLYRED:def 9</a></i>;<br/>
<b>consider </b><font color="Maroon">u</font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E15:25_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">p</font> <a href="polyred.html#R4">reduces_to</a> <font color="Maroon">q</font>,<font color="Maroon">u</font>,<font color="Maroon">T</font> )
 <b>by </b><i>, <a class="ref" href="polyred.html#D7">POLYRED:def 7</a></i>;<br/>
<b>consider </b><font color="Maroon">b</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font><b> such that </b><br/><a NAME="E17:25_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">p</font> <a href="polyred.html#R3">reduces_to</a> <font color="Maroon">q</font>,<font color="Maroon">u</font>,<font color="Maroon">b</font>,<font color="Maroon">T</font>
 <b>by </b><i>, <a class="ref" href="polyred.html#D6">POLYRED:def 6</a></i>;<br/>
<a NAME="E18:25_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</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="polyred.html#D5">POLYRED:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">u</font> = <font color="Maroon">u</font> as  <a href="polynom7.html#V1">non-zero</a> <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> <b>by </b><i><a class="ref" href="polynom7.html#D2">POLYNOM7:def 2</a></i>;<br/>
<a NAME="E20:25_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">q</font> <a href="polyred.html#R2">&lt;</a> <font color="Maroon">p</font>,<font color="Maroon">T</font>
 
<b>by </b><i>, <a class="ref" href="polyred.html#T43">POLYRED:43</a></i>;<br/>
<b>consider </b><font color="Maroon">m</font> being   <a href="polynom7.html#NM1">Monomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E22:25_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="polynom1.html#K25">-</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">u</font></span>)</span>
 <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">uu</font> = <font color="Maroon">u</font>, <font color="Maroon">pp</font> = <font color="Maroon">p</font>, <font color="Maroon">mm</font> = <font color="Maroon">m</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>
<b>reconsider </b><font color="Maroon">uu</font> = <font color="Maroon">uu</font>, <font color="Maroon">pp</font> = <font color="Maroon">pp</font>, <font color="Maroon">mm</font> = <font color="Maroon">mm</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span> ;<br/>
<a NAME="E25:25_1"/>
<font color="Maroon">uu</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="ideal_1.html#K7">-Ideal</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E26:25_1"/><b>then </b>
<font color="Maroon">mm</font> <a href="group_1.html#K10">*</a> <font color="Maroon">uu</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="ideal_1.html#K7">-Ideal</a> 
 
<b>by </b><i><a class="ref" href="ideal_1.html#D2">IDEAL_1:def 2</a></i>;<br/>
<a NAME="E27:25_1"/><b>then </b>
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">mm</font> <a href="group_1.html#K10">*</a> <font color="Maroon">uu</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="ideal_1.html#K7">-Ideal</a> 
 
<b>by </b><i><a class="ref" href="ideal_1.html#T13">IDEAL_1:13</a></i>;<br/>
<a NAME="E28:25_1"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">pp</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">mm</font> <a href="group_1.html#K10">*</a> <font color="Maroon">uu</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="ideal_1.html#K7">-Ideal</a> 
 
<b>by </b><i>, <a class="ref" href="ideal_1.html#D1">IDEAL_1:def 1</a></i>;<br/>
<a NAME="E29:25_1"/>
<font color="Maroon">mm</font> <a href="group_1.html#K10">*</a> <font color="Maroon">uu</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">u</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>
<a NAME="E30:25_1"/><b>then </b>
<font color="Maroon">p</font> <a href="polynom1.html#K25">-</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">u</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">pp</font> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">mm</font> <a href="group_1.html#K10">*</a> <font color="Maroon">uu</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E31:25_1"/><b>then </b><i><font color="Green">E76</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="ideal_1.html#K7">-Ideal</a> 
 
<b>by </b><i>, , <a class="ref" href="rlvect_1.html#D11">RLVECT_1:def 11</a></i>;<br/>
<a NAME="E32:25_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</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="polynom7.html#D2">POLYNOM7:def 2</a></i>;<br/>
<a NAME="E33:25_1"/><b>then </b>
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="polynom7.html#T1">POLYNOM7:1</a></i>;<br/>
<a NAME="E34:25_1"/><b>then </b><i><font color="Green">E78</font></i>: 
 <a href="termord.html#K3">HT</a> <font color="Maroon">p</font>,<font color="Maroon">T</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="termord.html#D6">TERMORD:def 6</a></i>;<br/>
<b>consider </b><font color="Maroon">b</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font><b> such that </b><br/><a NAME="E36:25_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">p</font> <a href="polyred.html#R3">reduces_to</a> <font color="Maroon">q</font>,<font color="Maroon">u</font>,<font color="Maroon">b</font>,<font color="Maroon">T</font>
 <b>by </b><i>, <a class="ref" href="polyred.html#D6">POLYRED:def 6</a></i>;<br/>
<a NAME="E37:25_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="polyred.html#D5">POLYRED:def 5</a></i>;<br/>
<a NAME="E38:25_1"/><b>then </b><i><font color="Green">E80</font></i>: 
<font color="Maroon">b</font> <a href="termord.html#R1">&lt;=</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">p</font>,<font color="Maroon">T</font>,<font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="termord.html#D6">TERMORD:def 6</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_1_2"/>
<font color="Maroon">b</font> <a href="pboole.html#R6">=</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">p</font>,<font color="Maroon">T</font>
 ;<br/><a NAME="E2:25_1_2"/><b>then </b>
<font color="Maroon">p</font> <a href="polyred.html#R8">top_reduces_to</a> <font color="Maroon">q</font>,<font color="Maroon">u</font>,<font color="Maroon">T</font>
 <b>by </b><i>, <a class="ref" href="polyred.html#D10">POLYRED:def 10</a></i>;<br/><a NAME="E3:25_1_2"/><b>then </b>
<font color="Maroon">p</font> <a href="polyred.html#R9">is_top_reducible_wrt</a> <font color="Maroon">u</font>,<font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="polyred.html#D11">POLYRED:def 11</a></i>;<br/><b>hence </b><a NAME="E4:25_1_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="polyred.html#D12">POLYRED:def 12</a></i>;<br/></div><b>end;</b></div>
<a NAME="E40:25_1"/><b>then </b>
<font color="Maroon">b</font> <a href="termord.html#R2">&lt;</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">p</font>,<font color="Maroon">T</font>,<font color="Maroon">T</font>
 
<b>by </b><i>, <a class="ref" href="termord.html#D3">TERMORD:def 3</a></i>;<br/>
<a NAME="E41:25_1"/><b>then </b><i><font color="Green">E81</font></i>: 
 <a href="termord.html#K3">HT</a> <font color="Maroon">p</font>,<font color="Maroon">T</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">q</font>
 
<b>by </b><i>, , <a class="ref" href="polyred.html#T40">POLYRED:40</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_1_3_1"/>
( <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> or <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_3_1_1"><b>case </b></a><a NAME="E1:25_1_3_1_1"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">q</font> = <font color="Maroon">q</font> as  <a href="polynom7.html#V1">non-zero</a> <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> <b>by </b><i><a class="ref" href="polynom7.html#D2">POLYNOM7:def 2</a></i>;<br/><a NAME="E3:25_1_3_1_1"/>
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="polynom7.html#T1">POLYNOM7:1</a></i>;<br/><a NAME="E4:25_1_3_1_1"/><b>then </b><i><font color="Green">E93</font></i>: 
 <a href="termord.html#K3">HT</a> <font color="Maroon">q</font>,<font color="Maroon">T</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="termord.html#D6">TERMORD:def 6</a></i>;<br/><a NAME="E5:25_1_3_1_1"/><i><font color="Green">E95</font></i>: 
 <a href="termord.html#K3">HT</a> <font color="Maroon">p</font>,<font color="Maroon">T</font> <a href="termord.html#R1">&lt;=</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">q</font>,<font color="Maroon">T</font>,<font color="Maroon">T</font>
 <b>by </b><i>, <a class="ref" href="termord.html#D6">TERMORD:def 6</a></i>;<br/><a NAME="E6:25_1_3_1_1"/>
 <a href="termord.html#K3">HT</a> <font color="Maroon">q</font>,<font color="Maroon">T</font> <a href="termord.html#R1">&lt;=</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">p</font>,<font color="Maroon">T</font>,<font color="Maroon">T</font>
 <b>by </b><i>, <a class="txt" href="groeb_1.html#E1:25"><i><font color="Green">E40</font></i></a>, <a class="ref" href="polyred.html#T42">POLYRED:42</a></i>;<br/><a NAME="E7:25_1_3_1_1"/><b>then </b><i><font color="Green">E97</font></i>: 
 <a href="termord.html#K3">HT</a> <font color="Maroon">q</font>,<font color="Maroon">T</font> <a href="pboole.html#R6">=</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">p</font>,<font color="Maroon">T</font>
 <b>by </b><i>, <a class="ref" href="termord.html#T7">TERMORD:7</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_3_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_1_3_1_1_1"/>
not <font color="Maroon">q</font> <a href="polyred.html#R10">is_top_reducible_wrt</a> <font color="Maroon">P</font>,<font color="Maroon">T</font>
 ;<br/><a NAME="E2:25_1_3_1_1_1"/><b>then </b>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font>
 <b>by </b><i/>;<br/><a NAME="E3:25_1_3_1_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="polyred.html#R1">&lt;=</a> <font color="Maroon">q</font>,<font color="Maroon">T</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:25_1_3_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="polyred.html#T29">POLYRED:29</a></i>;<br/></div><b>end;</b></div><b>then consider </b><font color="Maroon">u'</font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E10:25_1_3_1_1"/><i><font color="Green">E98</font></i>: 
( <font color="Maroon">u'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">q</font> <a href="polyred.html#R9">is_top_reducible_wrt</a> <font color="Maroon">u'</font>,<font color="Maroon">T</font> )
 <b>by </b><i><a class="ref" href="polyred.html#D12">POLYRED:def 12</a></i>;<br/><b>consider </b><font color="Maroon">q'</font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E12:25_1_3_1_1"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">q</font> <a href="polyred.html#R8">top_reduces_to</a> <font color="Maroon">q'</font>,<font color="Maroon">u'</font>,<font color="Maroon">T</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E3:25_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="polyred.html#D11">POLYRED:def 11</a></i>;<br/><a NAME="E13:25_1_3_1_1"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">q</font> <a href="polyred.html#R3">reduces_to</a> <font color="Maroon">q'</font>,<font color="Maroon">u'</font>, <a href="termord.html#K3">HT</a> <font color="Maroon">q</font>,<font color="Maroon">T</font>,<font color="Maroon">T</font>
 <b>by </b><i>, <a class="ref" href="polyred.html#D10">POLYRED:def 10</a></i>;<br/><a NAME="E14:25_1_3_1_1"/><b>then </b><i><font color="Green">E101</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> &amp; <font color="Maroon">u'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> &amp;  <a href="termord.html#K3">HT</a> <font color="Maroon">q</font>,<font color="Maroon">T</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">q</font> &amp;  ex <font color="Olive">s</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> st <br/>( <font color="Olive">s</font> <a href="polynom1.html#K9">+</a> <span class="p1">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">u'</font>,<font color="Maroon">T</font></span>)</span> <a href="pboole.html#R6">=</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">q</font>,<font color="Maroon">T</font> &amp; <font color="Maroon">q'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="polynom1.html#K25">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="polynom1.html#K15">.</a> <span class="p4">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">q</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p3">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">u'</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="polynom7.html#K5">*</a> <span class="p2">(<span class="default"><font color="Olive">s</font> <a href="polyred.html#K1">*'</a> <font color="Maroon">u'</font></span>)</span></span>)</span> ) )
 <b>by </b><i><a class="ref" href="polyred.html#D5">POLYRED:def 5</a></i>;<br/><b>consider </b><font color="Maroon">s</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font><b> such that </b><br/><a NAME="E16:25_1_3_1_1"/><i><font color="Green">E102</font></i>: 
( <font color="Maroon">s</font> <a href="polynom1.html#K9">+</a> <span class="p1">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">u'</font>,<font color="Maroon">T</font></span>)</span> <a href="pboole.html#R6">=</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">q</font>,<font color="Maroon">T</font> &amp; <font color="Maroon">q'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="polynom1.html#K25">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="polynom1.html#K15">.</a> <span class="p4">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">q</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p3">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">u'</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="polynom7.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="polyred.html#K1">*'</a> <font color="Maroon">u'</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E7:25_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="polyred.html#D5">POLYRED:def 5</a></i>;<br/><b>set </b><font color="Maroon">qq</font> = <font color="Maroon">p</font> <a href="polynom1.html#K25">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15">.</a> <span class="p4">(<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="vectsp_1.html#K5">/</a> <span class="p3">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">u'</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="polynom7.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="polyred.html#K1">*'</a> <font color="Maroon">u'</font></span>)</span></span>)</span>;<br/><a NAME="E17:25_1_3_1_1"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</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="polynom7.html#D2">POLYNOM7:def 2</a></i>;<br/><a NAME="E18:25_1_3_1_1"/><b>then </b>
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="polynom7.html#T1">POLYNOM7:1</a></i>;<br/><a NAME="E19:25_1_3_1_1"/><b>then </b>
 <a href="termord.html#K3">HT</a> <font color="Maroon">p</font>,<font color="Maroon">T</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="termord.html#D6">TERMORD:def 6</a></i>;<br/><a NAME="E20:25_1_3_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="polyred.html#R3">reduces_to</a> <font color="Maroon">p</font> <a href="polynom1.html#K25">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15">.</a> <span class="p4">(<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="vectsp_1.html#K5">/</a> <span class="p3">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">u'</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="polynom7.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="polyred.html#K1">*'</a> <font color="Maroon">u'</font></span>)</span></span>)</span>,<font color="Maroon">u'</font>, <a href="termord.html#K3">HT</a> <font color="Maroon">p</font>,<font color="Maroon">T</font>,<font color="Maroon">T</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:25_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="groeb_1.html#E9:25_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="groeb_1.html#E13:25_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="groeb_1.html#E15:25_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="polyred.html#D5">POLYRED:def 5</a></i>;<br/><a NAME="E21:25_1_3_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="polyred.html#R8">top_reduces_to</a> <font color="Maroon">p</font> <a href="polynom1.html#K25">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15">.</a> <span class="p4">(<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="vectsp_1.html#K5">/</a> <span class="p3">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">u'</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="polynom7.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="polyred.html#K1">*'</a> <font color="Maroon">u'</font></span>)</span></span>)</span>,<font color="Maroon">u'</font>,<font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="polyred.html#D10">POLYRED:def 10</a></i>;<br/><a NAME="E22:25_1_3_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="polyred.html#R9">is_top_reducible_wrt</a> <font color="Maroon">u'</font>,<font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="polyred.html#D11">POLYRED:def 11</a></i>;<br/><b>hence </b><a NAME="E23:25_1_3_1_1"/>
contradiction
 <b>by </b><i>, <a class="txt" href="groeb_1.html#E3:25_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="polyred.html#D12">POLYRED:def 12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_3_1_2"><b>case </b></a><a NAME="E1:25_1_3_1_2"/>
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font>
 ;<br/><div class="add"><b>then </b><i><font color="Green">E104</font></i>: <font color="Maroon">m</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">u</font> = 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K25">-</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">u</font></span>)</span></span>)</span> <a href="polynom1.html#K23">+</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">u</font></span>)</span>
<b>by </b><i>, <a class="ref" href="polyred.html#T2">POLYRED:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K23">+</a> <span class="p2">(<span class="default"><a href="polynom1.html#K24">-</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">u</font></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K23">+</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">u</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D23">POLYNOM1:def 23</a></i>
<br/>.= 
<font color="Maroon">p</font> <a href="polynom1.html#K23">+</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"><font color="Maroon">m</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">u</font></span>)</span></span>)</span> <a href="polynom1.html#K23">+</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">u</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#T80">POLYNOM1:80</a></i>
<br/>.= 
<font color="Maroon">p</font> <a href="polynom1.html#K23">+</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="polyred.html#T3">POLYRED:3</a></i>
<br/>.= 
<font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="polynom1.html#T82">POLYNOM1:82</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_3_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_1_3_1_2_2"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font>
 ;<br/><a NAME="E2:25_1_3_1_2_2"/>
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</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="polynom7.html#D2">POLYNOM7:def 2</a></i>;<br/><b>hence </b><a NAME="E3:25_1_3_1_2_2"/>
contradiction
 <b>by </b><i>, <a class="txt" href="groeb_1.html#E1:25"><i><font color="Green">E40</font></i></a>, <a class="ref" href="polyred.html#T5">POLYRED:5</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">m</font> = <font color="Maroon">m</font> as  <a href="polynom7.html#V1">non-zero</a> <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> <b>by </b><i><a class="ref" href="polynom7.html#D2">POLYNOM7:def 2</a></i>;<br/><a NAME="E5:25_1_3_1_2"/><i><font color="Green">E107</font></i>: 
 <a href="termord.html#K3">HT</a> <font color="Maroon">p</font>,<font color="Maroon">T</font> <a href="pboole.html#R6">=</a> <span class="p1">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m</font>,<font color="Maroon">T</font></span>)</span> <a href="polynom1.html#K9">+</a> <span class="p1">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">u</font>,<font color="Maroon">T</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="termord.html#T31">TERMORD:31</a></i>;<br/><b>set </b><font color="Maroon">pp</font> = <font color="Maroon">p</font> <a href="polynom1.html#K25">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15">.</a> <span class="p4">(<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="vectsp_1.html#K5">/</a> <span class="p3">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">u</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="polynom7.html#K5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m</font>,<font color="Maroon">T</font></span>)</span> <a href="polyred.html#K1">*'</a> <font color="Maroon">u</font></span>)</span></span>)</span>;<br/><a NAME="E6:25_1_3_1_2"/>
<font color="Maroon">p</font> <a href="polyred.html#R3">reduces_to</a> <font color="Maroon">p</font> <a href="polynom1.html#K25">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15">.</a> <span class="p4">(<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="vectsp_1.html#K5">/</a> <span class="p3">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">u</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="polynom7.html#K5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m</font>,<font color="Maroon">T</font></span>)</span> <a href="polyred.html#K1">*'</a> <font color="Maroon">u</font></span>)</span></span>)</span>,<font color="Maroon">u</font>, <a href="termord.html#K3">HT</a> <font color="Maroon">p</font>,<font color="Maroon">T</font>,<font color="Maroon">T</font>
 <b>by </b><i>, <a class="ref" href="groeb_1.html#T1" target="_self">Th1</a>, , <a class="txt" href="groeb_1.html#E1:25"><i><font color="Green">E40</font></i></a>, <a class="ref" href="polyred.html#D5">POLYRED:def 5</a></i>;<br/><a NAME="E7:25_1_3_1_2"/><b>then </b>
<font color="Maroon">p</font> <a href="polyred.html#R8">top_reduces_to</a> <font color="Maroon">p</font> <a href="polynom1.html#K25">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15">.</a> <span class="p4">(<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="vectsp_1.html#K5">/</a> <span class="p3">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">u</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="polynom7.html#K5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m</font>,<font color="Maroon">T</font></span>)</span> <a href="polyred.html#K1">*'</a> <font color="Maroon">u</font></span>)</span></span>)</span>,<font color="Maroon">u</font>,<font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="polyred.html#D10">POLYRED:def 10</a></i>;<br/><a NAME="E8:25_1_3_1_2"/><b>then </b>
<font color="Maroon">p</font> <a href="polyred.html#R9">is_top_reducible_wrt</a> <font color="Maroon">u</font>,<font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="polyred.html#D11">POLYRED:def 11</a></i>;<br/><b>hence </b><a NAME="E9:25_1_3_1_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="polyred.html#D12">POLYRED:def 12</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E43:25_1"/>
contradiction
 ;<br/>


</div><b>end;</b></div>


</div>
