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<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#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="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> <a href="algstr_0.html#V33" title="ALGSTR_0:attr.33">almost_left_invertible</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a> <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c4">f</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>let </b><font color="Maroon" title="c5">p</font> be  <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a> <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/>









<div><i><font color="Green" title="E52">A1</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" title="c4">f</font> <a href="polyred.html#R6" title="POLYRED:pred.6">is_reducible_wrt</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">T</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c6">g</font> being   <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><b> such that </b><br/><a NAME="E3:68_1"/><i><font color="Green" title="E53">A2</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="polyred.html#R4" title="POLYRED:pred.4">reduces_to</a> <font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="ref" href="polyred.html#D8" target="_self" title="POLYRED:def.8">Def8</a></i>;<br/><b>consider </b><font color="Maroon" title="c7">b</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E5:68_1"/><i><font color="Green" title="E54">A3</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="polyred.html#R3" title="POLYRED:pred.3">reduces_to</a> <font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="txt" href="polyred.html#E3:68_1"><i><font color="Green" title="E53">A2</font></i></a>, <a class="ref" href="polyred.html#D6" target="_self" title="POLYRED:def.6">Def6</a></i>;<br/><a NAME="E6:68_1"/><i><font color="Green" title="E55">A4</font></i>: 
( <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c4">f</font> &amp;  ex <font color="Olive" title="b1">s</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font> st <br/>( <font color="Olive" title="b1">s</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <span class="p1">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> &amp; <font color="Maroon" title="c6">g</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">f</font> <a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p3">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">s</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span> ) )
 <b>by </b><i><a class="txt" href="polyred.html#E5:68_1"><i><font color="Green" title="E54">A3</font></i></a>, <a class="ref" href="polyred.html#D5" target="_self" title="POLYRED:def.5">Def5</a></i>;<br/><a NAME="E7:68_1"/><b>then </b>
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">T</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c7">b</font>
 <b>by </b><i><a class="ref" href="termord.html#T1" title="TERMORD:th.1">TERMORD:1</a></i>;<br/><b>hence </b><a NAME="E8:68_1"/>
 ex <font color="Olive" title="b1">b</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font> st <br/>( <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c4">f</font> &amp;  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">T</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Olive" title="b1">b</font> )
 <b>by </b><i><a class="txt" href="polyred.html#E6:68_1"><i><font color="Green" title="E55">A4</font></i></a></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" title="b1">b</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font> st <br/>( <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c4">f</font> &amp;  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">T</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Olive" title="b1">b</font> )
 ;<br/><b>then consider </b><font color="Maroon" title="c6">b</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E3:68_2"/><i><font color="Green" title="E53">A5</font></i>: 
( <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c4">f</font> &amp;  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">T</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c6">b</font> )
 ;<br/><b>consider </b><font color="Maroon" title="c7">s</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E5:68_2"/><i><font color="Green" title="E54">A6</font></i>: 
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">T</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Maroon" title="c7">s</font>
 <b>by </b><i><a class="txt" href="polyred.html#E3:68_2"><i><font color="Green" title="E53">A5</font></i></a>, <a class="ref" href="termord.html#T1" title="TERMORD:th.1">TERMORD:1</a></i>;<br/><b>set </b><font color="Maroon" title="c8">g</font> = <font color="Maroon" title="c4">f</font> <a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p3">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">s</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span>;<br/><a NAME="E6:68_2"/>
( <font color="Maroon" title="c4">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> &amp; <font color="Maroon" title="c5">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> )
 <b>by </b><i><a class="txt" href="polyred.html#E3:68_2"><i><font color="Green" title="E53">A5</font></i></a>, <a class="ref" href="polynom7.html#T1" title="POLYNOM7:th.1">POLYNOM7:1</a>, <a class="ref" href="polynom7.html#D2" title="POLYNOM7:def.2">POLYNOM7:def 2</a></i>;<br/><a NAME="E7:68_2"/><b>then </b>
<font color="Maroon" title="c4">f</font> <a href="polyred.html#R3" title="POLYRED:pred.3">reduces_to</a> <font color="Maroon" title="c4">f</font> <a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p3">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">s</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="txt" href="polyred.html#E3:68_2"><i><font color="Green" title="E53">A5</font></i></a>, <a class="txt" href="polyred.html#E5:68_2"><i><font color="Green" title="E54">A6</font></i></a>, <a class="ref" href="polyred.html#D5" target="_self" title="POLYRED:def.5">Def5</a></i>;<br/><a NAME="E8:68_2"/><b>then </b>
<font color="Maroon" title="c4">f</font> <a href="polyred.html#R4" title="POLYRED:pred.4">reduces_to</a> <font color="Maroon" title="c4">f</font> <a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p3">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">s</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="ref" href="polyred.html#D6" target="_self" title="POLYRED:def.6">Def6</a></i>;<br/><b>hence </b><a NAME="E9:68_2"/>
<font color="Maroon" title="c4">f</font> <a href="polyred.html#R6" title="POLYRED:pred.6">is_reducible_wrt</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="ref" href="polyred.html#D8" target="_self" title="POLYRED:def.8">Def8</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:68"/>
( <font color="Maroon" title="c4">f</font> <a href="polyred.html#R6" title="POLYRED:pred.6">is_reducible_wrt</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">T</font> iff  ex <font color="Olive" title="b1">b</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font> st <br/>( <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c4">f</font> &amp;  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">T</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Olive" title="b1">b</font> ) )
 <b>by </b><i><a class="txt" href="polyred.html#E1:68"><i><font color="Green" title="E52">A1</font></i></a></i>;<br/>


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