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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#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> <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="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>let </b><font color="Maroon">P</font> be  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>;<br/>









<b>assume </b><a NAME="E1:62"/>
<font color="Maroon">f</font> <a href="groeb_2.html#R6" target="_self">has_a_Standard_Representation_of</a> <font color="Maroon">P</font>,<font color="Maroon">T</font>
 ;<br/>

<b>then consider </b><font color="Maroon">A</font> being    <a href="ideal_1.html#M2">LeftLinearCombination</a> of <font color="Maroon">P</font><b> such that </b><br/><a NAME="E3:62"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">A</font> <a href="groeb_2.html#R4" target="_self">is_Standard_Representation_of</a> <font color="Maroon">f</font>,<font color="Maroon">P</font>,<font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="groeb_2.html#T2" target="_self">Th2</a></i>;<br/>
<b>consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">m</font> being  <a href="polynom7.html#V1">non-zero</a> <a href="polynom7.html#NM1">Monomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>, <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="E5:62"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">p</font> &amp;  <a href="termord.html#K3">HT</a> <font color="Maroon">f</font>,<font color="Maroon">T</font> <a href="pboole.html#R6">=</a>  <a href="termord.html#K3">HT</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">T</font> )
 <b>by </b><i><a class="txt" href="groeb_2.html#E5:62"><i><font color="Green">E33</font></i></a>, </i>;<br/>
<a NAME="E6:62"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">A</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> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">A</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">p</font> )
 
<b>by </b><i><a class="txt" href="groeb_2.html#E6:62"><i><font color="Green">E34</font></i></a>, <a class="ref" href="polynom7.html#D2">POLYNOM7:def 2</a></i>;<br/>
<a NAME="E7:62"/><i><font color="Green">E35</font></i>: 
<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>
 
<b>by </b><i><a class="ref" href="polynom7.html#D2">POLYNOM7:def 2</a></i>;<br/>
<a NAME="E8:62"/><b>then </b>
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">f</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="E9:62"/><b>then </b><i><font color="Green">E37</font></i>: 
 <a href="termord.html#K3">HT</a> <font color="Maroon">f</font>,<font color="Maroon">T</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="termord.html#D6">TERMORD:def 6</a></i>;<br/>
<b>set </b><font color="Maroon">s</font> =  <a href="termord.html#K3">HT</a> <font color="Maroon">m</font>,<font color="Maroon">T</font>;<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> <span class="p4">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">f</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">p</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">p</font></span>)</span></span>)</span>;<br/>
<a NAME="E10:62"/>
 <a href="termord.html#K3">HT</a> <font color="Maroon">f</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">p</font>,<font color="Maroon">T</font></span>)</span>
 
<b>by </b><i><a class="txt" href="groeb_2.html#E6:62"><i><font color="Green">E34</font></i></a>, <a class="ref" href="termord.html#T31">TERMORD:31</a></i>;<br/>
<a NAME="E11:62"/><b>then </b>
<font color="Maroon">f</font> <a href="polyred.html#R3">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> <span class="p4">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">f</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">p</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">p</font></span>)</span></span>)</span>,<font color="Maroon">p</font>, <a href="termord.html#K3">HT</a> <font color="Maroon">f</font>,<font color="Maroon">T</font>,<font color="Maroon">T</font>
 
<b>by </b><i><a class="txt" href="groeb_2.html#E7:62"><i><font color="Green">E35</font></i></a>, , <a class="txt" href="groeb_2.html#E9:62"><i><font color="Green">E37</font></i></a>, <a class="ref" href="polyred.html#D5">POLYRED:def 5</a></i>;<br/>
<a NAME="E12:62"/><b>then </b>
<font color="Maroon">f</font> <a href="polyred.html#R8">top_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> <span class="p4">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">f</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">p</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">p</font></span>)</span></span>)</span>,<font color="Maroon">p</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="E13:62"/><b>then </b>
<font color="Maroon">f</font> <a href="polyred.html#R9">is_top_reducible_wrt</a> <font color="Maroon">p</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="E14:62"/>
<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>
 <b>by </b><i><a class="txt" href="groeb_2.html#E6:62"><i><font color="Green">E34</font></i></a>, <a class="ref" href="polyred.html#D12">POLYRED:def 12</a></i>;<br/>


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