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<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#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> <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="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="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="E1:26"/><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#R10">is_top_reducible_wrt</a> <font color="Maroon">P</font>,<font color="Maroon">T</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_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="E1:26_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="groeb_1.html#K2" target="_self">HT</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="ideal_1.html#K7">-Ideal</a> </span>)</span>,<font color="Maroon">T</font>
 ;<br/><a NAME="E2:26_1"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Olive">p</font>,<font color="Maroon">T</font></span>)</span> where <font color="Olive">p</font> is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> : ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="ideal_1.html#K7">-Ideal</a>  &amp; <font color="Olive">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> ) </span>}</span> 
 ;<br/><b>then 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="E4:26_1"/><i><font color="Green">E42</font></i>: 
( <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> &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; <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> )
 ;<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>, <a class="ref" href="polynom7.html#D2">POLYNOM7:def 2</a></i>;<br/><a NAME="E6:26_1"/>
<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>, </i>;<br/><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="E8:26_1"/><i><font color="Green">E43</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#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="E10:26_1"/><i><font color="Green">E44</font></i>: 
<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#D11">POLYRED:def 11</a></i>;<br/><a NAME="E11:26_1"/><i><font color="Green">E45</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>, <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="polyred.html#D10">POLYRED:def 10</a></i>;<br/><b>then 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="E13:26_1"/><i><font color="Green">E46</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">p</font>,<font color="Maroon">T</font> &amp; <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"><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> )
 <b>by </b><i><a class="ref" href="polyred.html#D5">POLYRED:def 5</a></i>;<br/><a NAME="E14:26_1"/>
<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/><a NAME="E15:26_1"/><b>then </b>
 <a href="termord.html#K3">HT</a> <font color="Maroon">u</font>,<font color="Maroon">T</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Olive">r</font>,<font color="Maroon">T</font></span>)</span> where <font color="Olive">r</font> is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> : ( <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">r</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> ) </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E16:26_1"/><b>then </b><i><font color="Green">E47</font></i>: 
 <a href="termord.html#K3">HT</a> <font color="Maroon">u</font>,<font color="Maroon">T</font> <a href="hidden.html#R2">in</a>  <a href="groeb_1.html#K2" target="_self">HT</a> <font color="Maroon">P</font>,<font color="Maroon">T</font>
 ;<br/><a NAME="E17:26_1"/>
 <a href="termord.html#K3">HT</a> <font color="Maroon">u</font>,<font color="Maroon">T</font> <a href="polynom1.html#R3">divides</a> <font color="Maroon">b</font>
 <b>by </b><i>, , <a class="ref" href="polynom1.html#T54">POLYNOM1:54</a></i>;<br/><b>hence </b><a NAME="E18:26_1"/>
 ex <font color="Olive">b'</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> st <br/>( <font color="Olive">b'</font> <a href="hidden.html#R2">in</a>  <a href="groeb_1.html#K2" target="_self">HT</a> <font color="Maroon">P</font>,<font color="Maroon">T</font> &amp; <font color="Olive">b'</font> <a href="polynom1.html#R3">divides</a> <font color="Maroon">b</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:26"/>
for <font color="Olive">b</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>  st <font color="Olive">b</font> <a href="hidden.html#R2">in</a>  <a href="groeb_1.html#K2" target="_self">HT</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="ideal_1.html#K7">-Ideal</a> </span>)</span>,<font color="Maroon">T</font> holds <br/> ex <font color="Olive">b'</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> st <br/>( <font color="Olive">b'</font> <a href="hidden.html#R2">in</a>  <a href="groeb_1.html#K2" target="_self">HT</a> <font color="Maroon">P</font>,<font color="Maroon">T</font> &amp; <font color="Olive">b'</font> <a href="polynom1.html#R3">divides</a> <font color="Olive">b</font> )
 ;<br/>


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