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









<div><i><font color="Green">E23</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:68_1"/>
<font color="Maroon">f</font> <a href="polyred.html#R6" target="_self">is_reducible_wrt</a> <font color="Maroon">p</font>,<font color="Maroon">T</font>
 ;<br/><b>then consider </b><font color="Maroon">g</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="E3:68_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">f</font> <a href="polyred.html#R4" target="_self">reduces_to</a> <font color="Maroon">g</font>,<font color="Maroon">p</font>,<font color="Maroon">T</font>
 <b>by </b><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="E5:68_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">f</font> <a href="polyred.html#R3" target="_self">reduces_to</a> <font color="Maroon">g</font>,<font color="Maroon">p</font>,<font color="Maroon">b</font>,<font color="Maroon">T</font>
 <b>by </b><i>, </i>;<br/><a NAME="E6:68_1"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">f</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">p</font>,<font color="Maroon">T</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">g</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">f</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">f</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></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">p</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" target="_self">*'</a> <font color="Maroon">p</font></span>)</span></span>)</span> ) )
 <b>by </b><i>, </i>;<br/><a NAME="E7:68_1"/><b>then </b>
 <a href="termord.html#K3">HT</a> <font color="Maroon">p</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="termord.html#T1">TERMORD:1</a></i>;<br/><b>hence </b><a NAME="E8:68_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="polynom1.html#K12">Support</a> <font color="Maroon">f</font> &amp;  <a href="termord.html#K3">HT</a> <font color="Maroon">p</font>,<font color="Maroon">T</font> <a href="polynom1.html#R3">divides</a> <font color="Olive">b</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:68_2"/>
 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="polynom1.html#K12">Support</a> <font color="Maroon">f</font> &amp;  <a href="termord.html#K3">HT</a> <font color="Maroon">p</font>,<font color="Maroon">T</font> <a href="polynom1.html#R3">divides</a> <font color="Olive">b</font> )
 ;<br/><b>then 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="E3:68_2"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">f</font> &amp;  <a href="termord.html#K3">HT</a> <font color="Maroon">p</font>,<font color="Maroon">T</font> <a href="polynom1.html#R3">divides</a> <font color="Maroon">b</font> )
 ;<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="E5:68_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">s</font>
 <b>by </b><i>, <a class="ref" href="termord.html#T1">TERMORD:1</a></i>;<br/><b>set </b><font color="Maroon">g</font> = <font color="Maroon">f</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">f</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></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">p</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" target="_self">*'</a> <font color="Maroon">p</font></span>)</span></span>)</span>;<br/><a NAME="E6:68_2"/>
( <font color="Maroon">f</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">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#T1">POLYNOM7:1</a>, <a class="ref" href="polynom7.html#D2">POLYNOM7:def 2</a></i>;<br/><a NAME="E7:68_2"/><b>then </b>
<font color="Maroon">f</font> <a href="polyred.html#R3" target="_self">reduces_to</a> <font color="Maroon">f</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">f</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></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">p</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" target="_self">*'</a> <font color="Maroon">p</font></span>)</span></span>)</span>,<font color="Maroon">p</font>,<font color="Maroon">b</font>,<font color="Maroon">T</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E8:68_2"/><b>then </b>
<font color="Maroon">f</font> <a href="polyred.html#R4" target="_self">reduces_to</a> <font color="Maroon">f</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">f</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></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">p</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" target="_self">*'</a> <font color="Maroon">p</font></span>)</span></span>)</span>,<font color="Maroon">p</font>,<font color="Maroon">T</font>
 <b>by </b><i><a class="txt" href="polyred.html#E5:68_2"><i><font color="Green">E49</font></i></a></i>;<br/><b>hence </b><a NAME="E9:68_2"/>
<font color="Maroon">f</font> <a href="polyred.html#R6" target="_self">is_reducible_wrt</a> <font color="Maroon">p</font>,<font color="Maroon">T</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:68"/>
( <font color="Maroon">f</font> <a href="polyred.html#R6" target="_self">is_reducible_wrt</a> <font color="Maroon">p</font>,<font color="Maroon">T</font> iff  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="polynom1.html#K12">Support</a> <font color="Maroon">f</font> &amp;  <a href="termord.html#K3">HT</a> <font color="Maroon">p</font>,<font color="Maroon">T</font> <a href="polynom1.html#R3">divides</a> <font color="Olive">b</font> ) )
 <b>by </b><i/>;<br/>


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