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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#V2" title="BAGORDER:attr.2">admissible</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="rlvect_1.html#V4" title="RLVECT_1:attr.4">add-associative</a> <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">right_zeroed</a> <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">right_complementable</a> <a href="group_1.html#V2" title="GROUP_1:attr.2">unital</a> <a href="vectsp_1.html#V8" title="VECTSP_1:attr.8">distributive</a> <a href="vectsp_2.html#V1" title="VECTSP_2:attr.1">domRing-like</a>  <a href="vectsp_1.html#L3" title="VECTSP_1:struct.3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c4">p</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">m</font> be  <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a> <a href="polynom7.html#NM1" title="POLYNOM7:NM.1">Monomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>;<br/><b>let </b><font color="Maroon" title="c6">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>











<b>assume </b><a NAME="E1:54"/><i><font color="Green" title="E45">A1</font></i>: 
<font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">p</font></span>)</span>
 ;<br/>

<a NAME="E2:54"/><b>then </b><i><font color="Green" title="E46">A2</font></i>: 
<font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="groeb_3.html#T10" target="_self" title="GROEB_3:th.10">Th10</a></i>;<br/>
<b>set </b><font color="Maroon" title="c7">l</font> =  <a href="groeb_3.html#K6" title="GROEB_3:func.6">Low</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font>;<br/>
<b>set </b><font color="Maroon" title="c8">lm</font> =  <a href="groeb_3.html#K6" title="GROEB_3:func.6">Low</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font>;<br/>
<i><font color="Green" title="E47">A3</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <span class="p2">(<span class="default"><a href="groeb_3.html#K6" title="GROEB_3:func.6">Low</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span></span>)</span> = 
 <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K6" title="GROEB_3:func.6">Low</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
;<br/>
<a NAME="E4:54"/><i><font color="Green" title="E48">A4</font></i>: 
 <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <span class="p2">(<span class="default"><a href="groeb_3.html#K6" title="GROEB_3:func.6">Low</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">s</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Olive" title="b2">t</font></span>)</span> where <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</font> is    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font> : ( <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c5">m</font> &amp; <font color="Olive" title="b2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K6" title="GROEB_3:func.6">Low</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="termord.html#T30" title="TERMORD:th.30">TERMORD:30</a></i>;<br/>
<a NAME="E5:54"/><i><font color="Green" title="E49">A5</font></i>: 
 <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">s</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Olive" title="b2">t</font></span>)</span> where <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</font> is    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font> : ( <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c5">m</font> &amp; <font color="Olive" title="b2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">p</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="termord.html#T30" title="TERMORD:th.30">TERMORD:30</a></i>;<br/>
<div><i><font color="Green" title="E50">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:54_2"/><i><font color="Green" title="E51">A7</font></i>: 
<font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <span class="p2">(<span class="default"><a href="groeb_3.html#K6" title="GROEB_3:func.6">Low</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c10">u'</font> = <font color="Maroon" title="c9">u</font> as    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font> ;<br/><a NAME="E3:54_2"/>
<font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">s</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Olive" title="b2">t</font></span>)</span> where <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</font> is    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font> : ( <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c5">m</font> &amp; <font color="Olive" title="b2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K6" title="GROEB_3:func.6">Low</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="groeb_3.html#E4:54"><i><font color="Green" title="E48">A4</font></i></a>, <a class="txt" href="groeb_3.html#E1:54_2"><i><font color="Green" title="E51">A7</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">s</font>, <font color="Maroon" title="c12">t</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E5:54_2"/><i><font color="Green" title="E52">A8</font></i>: 
( <font color="Maroon" title="c10">u'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">s</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c12">t</font> &amp; <font color="Maroon" title="c11">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c5">m</font> &amp; <font color="Maroon" title="c12">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K6" title="GROEB_3:func.6">Low</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span> )
 ;<br/><a NAME="E6:54_2"/><i><font color="Green" title="E53">A9</font></i>: 
<span class="p1">(<span class="default"><a href="groeb_3.html#K6" title="GROEB_3:func.6">Low</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c12">t</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">L</font>
 <b>by </b><i><a class="txt" href="groeb_3.html#E5:54_2"><i><font color="Green" title="E52">A8</font></i></a>, <a class="ref" href="polynom1.html#D9" title="POLYNOM1:def.9">POLYNOM1:def 9</a></i>;<br/><a NAME="E7:54_2"/>
<font color="Maroon" title="c5">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K26" title="POLYNOM1:func.26">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>
 <b>by </b><i><a class="ref" href="polynom7.html#D2" title="POLYNOM7:def.2">POLYNOM7:def 2</a></i>;<br/><a NAME="E8:54_2"/><b>then </b>
 <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c5">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="polynom7.html#T1" title="POLYNOM7:th.1">POLYNOM7:1</a></i>;<br/><a NAME="E9:54_2"/><b>then </b><i><font color="Green" title="E54">A10</font></i>: 
 <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c5">m</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="polynom7.html#T7" title="POLYNOM7:th.7">POLYNOM7:7</a></i>;<br/><a NAME="E10:54_2"/><b>then </b>
 <a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c5">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c5">m</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E11:54_2"/><b>then </b><i><font color="Green" title="E55">A11</font></i>: 
<font color="Maroon" title="c5">m</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <span class="p1">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">L</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D9" title="POLYNOM1:def.9">POLYNOM1:def 9</a></i>;<br/><a NAME="E12:54_2"/><i><font color="Green" title="E56">A12</font></i>: 
 <a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">s</font>
 <b>by </b><i><a class="txt" href="groeb_3.html#E5:54_2"><i><font color="Green" title="E52">A8</font></i></a>, <a class="txt" href="groeb_3.html#E9:54_2"><i><font color="Green" title="E54">A10</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c10">u'</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <span class="p2">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c12">t</font></span>)</span>
<b>by </b><i><a class="txt" href="groeb_3.html#E5:54_2"><i><font color="Green" title="E52">A8</font></i></a>, <a class="ref" href="polyred.html#T7" title="POLYRED:th.7">POLYRED:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <span class="p2">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="groeb_3.html#K6" title="GROEB_3:func.6">Low</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c12">t</font></span>)</span>
<b>by </b><i><a class="txt" href="groeb_3.html#E1:54"><i><font color="Green" title="E45">A1</font></i></a>, <a class="txt" href="groeb_3.html#E5:54_2"><i><font color="Green" title="E52">A8</font></i></a>, <a class="ref" href="groeb_3.html#T31" target="_self" title="GROEB_3:th.31">Th31</a></i>
;<br/><a NAME="E14:54_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c10">u'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">L</font>
 <b>by </b><i><a class="txt" href="groeb_3.html#E6:54_2"><i><font color="Green" title="E53">A9</font></i></a>, <a class="txt" href="groeb_3.html#E11:54_2"><i><font color="Green" title="E55">A11</font></i></a>, <a class="ref" href="vectsp_2.html#D5" title="VECTSP_2:def.5">VECTSP_2:def 5</a></i>;<br/><a NAME="E15:54_2"/><b>then </b><i><font color="Green" title="E57">A13</font></i>: 
<font color="Maroon" title="c10">u'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom1.html#D9" title="POLYNOM1:def.9">POLYNOM1:def 9</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:54_2_2"/>
not <font color="Maroon" title="c11">s</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c12">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K6" title="GROEB_3:func.6">Low</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span>
 ;<br/><a NAME="E2:54_2_2"/><b>then </b><i><font color="Green" title="E58">A14</font></i>: 
<font color="Maroon" title="c11">s</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c12">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" title="GROEB_3:func.5">Up</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_3.html#E15:54_2"><i><font color="Green" title="E57">A13</font></i></a>, <a class="txt" href="groeb_3.html#E5:54_2"><i><font color="Green" title="E52">A8</font></i></a>, <a class="txt" href="groeb_3.html#E2:54"><i><font color="Green" title="E46">A2</font></i></a>, <a class="ref" href="groeb_3.html#T28" target="_self" title="GROEB_3:th.28">Th28</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c13">t'</font> be   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>;<br/><b>assume </b><a NAME="E1:54_2_2_1"/>
<font color="Maroon" title="c13">t'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K6" title="GROEB_3:func.6">Low</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span>
 ;<br/><a NAME="E2:54_2_2_1"/><b>then </b>
<font color="Maroon" title="c11">s</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c13">t'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K6" title="GROEB_3:func.6">Low</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_3.html#E12:54_2"><i><font color="Green" title="E56">A12</font></i></a>, <a class="txt" href="groeb_3.html#E1:54"><i><font color="Green" title="E45">A1</font></i></a>, <a class="ref" href="groeb_3.html#T40" target="_self" title="GROEB_3:th.40">Th40</a></i>;<br/><a NAME="E3:54_2_2_1"/><b>then </b><i><font color="Green" title="E59">A15</font></i>: 
<font color="Maroon" title="c11">s</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c13">t'</font> <a href="termord.html#R2" title="TERMORD:pred.2">&lt;</a> <font color="Maroon" title="c11">s</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c12">t</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="txt" href="groeb_3.html#E2:54_2_2"><i><font color="Green" title="E58">A14</font></i></a>, <a class="txt" href="groeb_3.html#E2:54"><i><font color="Green" title="E46">A2</font></i></a>, <a class="ref" href="groeb_3.html#T29" target="_self" title="GROEB_3:th.29">Th29</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:54_2_2_1_1"/>
<font color="Maroon" title="c12">t</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a> <font color="Maroon" title="c13">t'</font>,<font color="Maroon" title="c2">T</font>
 ;<br/><a NAME="E2:54_2_2_1_1"/><b>then </b>
<font color="Maroon" title="c11">s</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c12">t</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a> <font color="Maroon" title="c11">s</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c13">t'</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="ref" href="groeb_3.html#T2" target="_self" title="GROEB_3:th.2">Th2</a></i>;<br/><b>hence </b><a NAME="E3:54_2_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="groeb_3.html#E3:54_2_2_1"><i><font color="Green" title="E59">A15</font></i></a>, <a class="ref" href="termord.html#T5" title="TERMORD:th.5">TERMORD:5</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:54_2_2_1"/>
<font color="Maroon" title="c13">t'</font> <a href="termord.html#R2" title="TERMORD:pred.2">&lt;</a> <font color="Maroon" title="c12">t</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="ref" href="termord.html#T5" title="TERMORD:th.5">TERMORD:5</a></i>;<br/></div><b>end;</b></div><a NAME="E4:54_2_2"/><b>then </b>
<font color="Maroon" title="c12">t</font> <a href="termord.html#R2" title="TERMORD:pred.2">&lt;</a> <font color="Maroon" title="c12">t</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="txt" href="groeb_3.html#E5:54_2"><i><font color="Green" title="E52">A8</font></i></a></i>;<br/><b>hence </b><a NAME="E5:54_2_2"/>
contradiction
 <b>by </b><i><a class="ref" href="termord.html#D3" title="TERMORD:def.3">TERMORD:def 3</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E17:54_2"/>
<font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K6" title="GROEB_3:func.6">Low</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_3.html#E5:54_2"><i><font color="Green" title="E52">A8</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:54_3"/><i><font color="Green" title="E51">A16</font></i>: 
<font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K6" title="GROEB_3:func.6">Low</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c10">u'</font> = <font color="Maroon" title="c9">u</font> as    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font> ;<br/><a NAME="E3:54_3"/>
 <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K6" title="GROEB_3:func.6">Low</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_3.html#E2:54"><i><font color="Green" title="E46">A2</font></i></a>, <a class="ref" href="groeb_3.html#T26" target="_self" title="GROEB_3:th.26">Th26</a></i>;<br/><a NAME="E4:54_3"/><b>then </b>
<font color="Maroon" title="c10">u'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_3.html#E1:54_3"><i><font color="Green" title="E51">A16</font></i></a></i>;<br/><a NAME="E5:54_3"/><b>then </b>
<font color="Maroon" title="c10">u'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">s</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Olive" title="b2">t</font></span>)</span> where <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</font> is    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font> : ( <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c5">m</font> &amp; <font color="Olive" title="b2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">p</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="groeb_3.html#E5:54"><i><font color="Green" title="E49">A5</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">s</font>, <font color="Maroon" title="c12">t</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E7:54_3"/><i><font color="Green" title="E52">A17</font></i>: 
( <font color="Maroon" title="c9">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">s</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c12">t</font> &amp; <font color="Maroon" title="c11">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c5">m</font> &amp; <font color="Maroon" title="c12">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">p</font> )
 ;<br/><a NAME="E8:54_3"/>
<font color="Maroon" title="c5">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K26" title="POLYNOM1:func.26">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>
 <b>by </b><i><a class="ref" href="polynom7.html#D2" title="POLYNOM7:def.2">POLYNOM7:def 2</a></i>;<br/><a NAME="E9:54_3"/><b>then </b>
 <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c5">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="polynom7.html#T1" title="POLYNOM7:th.1">POLYNOM7:1</a></i>;<br/><a NAME="E10:54_3"/><b>then </b><i><font color="Green" title="E53">A18</font></i>: 
 <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c5">m</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="polynom7.html#T7" title="POLYNOM7:th.7">POLYNOM7:7</a></i>;<br/><a NAME="E11:54_3"/><b>then </b>
 <a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c5">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c5">m</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E12:54_3"/><b>then </b><i><font color="Green" title="E54">A19</font></i>: 
<font color="Maroon" title="c5">m</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <span class="p1">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">L</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D9" title="POLYNOM1:def.9">POLYNOM1:def 9</a></i>;<br/><a NAME="E13:54_3"/><i><font color="Green" title="E55">A20</font></i>: 
 <a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">s</font>
 <b>by </b><i><a class="txt" href="groeb_3.html#E7:54_3"><i><font color="Green" title="E52">A17</font></i></a>, <a class="txt" href="groeb_3.html#E10:54_3"><i><font color="Green" title="E53">A18</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E14:54_3"/><i><font color="Green" title="E56">A21</font></i>: 
<font color="Maroon" title="c4">p</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c12">t</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">L</font>
 <b>by </b><i><a class="txt" href="groeb_3.html#E7:54_3"><i><font color="Green" title="E52">A17</font></i></a>, <a class="ref" href="polynom1.html#D9" title="POLYNOM1:def.9">POLYNOM1:def 9</a></i>;<br/><a NAME="E15:54_3"/><i><font color="Green" title="E57">A22</font></i>: 
<font color="Maroon" title="c12">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K6" title="GROEB_3:func.6">Low</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_3.html#E13:54_3"><i><font color="Green" title="E55">A20</font></i></a>, <a class="txt" href="groeb_3.html#E7:54_3"><i><font color="Green" title="E52">A17</font></i></a>, <a class="txt" href="groeb_3.html#E1:54_3"><i><font color="Green" title="E51">A16</font></i></a>, <a class="txt" href="groeb_3.html#E1:54"><i><font color="Green" title="E45">A1</font></i></a>, <a class="ref" href="groeb_3.html#T40" target="_self" title="GROEB_3:th.40">Th40</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <span class="p2">(<span class="default"><a href="groeb_3.html#K6" title="GROEB_3:func.6">Low</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c12">t</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <span class="p2">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="groeb_3.html#K6" title="GROEB_3:func.6">Low</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c12">t</font></span>)</span>
<b>by </b><i><a class="ref" href="polyred.html#T7" title="POLYRED:th.7">POLYRED:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <span class="p2">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c12">t</font></span>)</span>
<b>by </b><i><a class="txt" href="groeb_3.html#E15:54_3"><i><font color="Green" title="E57">A22</font></i></a>, <a class="txt" href="groeb_3.html#E1:54"><i><font color="Green" title="E45">A1</font></i></a>, <a class="ref" href="groeb_3.html#T31" target="_self" title="GROEB_3:th.31">Th31</a></i>
;<br/><a NAME="E17:54_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <span class="p2">(<span class="default"><a href="groeb_3.html#K6" title="GROEB_3:func.6">Low</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c10">u'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">L</font>
 <b>by </b><i><a class="txt" href="groeb_3.html#E7:54_3"><i><font color="Green" title="E52">A17</font></i></a>, <a class="txt" href="groeb_3.html#E14:54_3"><i><font color="Green" title="E56">A21</font></i></a>, <a class="txt" href="groeb_3.html#E13:54_3"><i><font color="Green" title="E55">A20</font></i></a>, <a class="txt" href="groeb_3.html#E12:54_3"><i><font color="Green" title="E54">A19</font></i></a>, <a class="ref" href="vectsp_2.html#D5" title="VECTSP_2:def.5">VECTSP_2:def 5</a></i>;<br/><b>hence </b><a NAME="E18:54_3"/>
<font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <span class="p2">(<span class="default"><a href="groeb_3.html#K6" title="GROEB_3:func.6">Low</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="polynom1.html#D9" title="POLYNOM1:def.9">POLYNOM1:def 9</a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:54"/><b>then </b><i><font color="Green" title="E51">A23</font></i>: 
 <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <span class="p2">(<span class="default"><a href="groeb_3.html#K6" title="GROEB_3:func.6">Low</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K6" title="GROEB_3:func.6">Low</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="groeb_3.html#E6:54"><i><font color="Green" title="E50">A6</font></i></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:54_4"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <span class="p2">(<span class="default"><a href="groeb_3.html#K6" title="GROEB_3:func.6">Low</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c10">b</font> = <font color="Maroon" title="c9">x</font> as    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:54_4_1_1"/>
( <font color="Maroon" title="c10">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <span class="p2">(<span class="default"><a href="groeb_3.html#K6" title="GROEB_3:func.6">Low</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span></span>)</span> or  not <font color="Maroon" title="c10">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <span class="p2">(<span class="default"><a href="groeb_3.html#K6" title="GROEB_3:func.6">Low</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_4_1_1_1"><b>case </b></a><a NAME="E1:54_4_1_1_1"/><i><font color="Green" title="E52">A24</font></i>: 
<font color="Maroon" title="c10">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <span class="p2">(<span class="default"><a href="groeb_3.html#K6" title="GROEB_3:func.6">Low</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:54_4_1_1_1"/><b>then </b><i><font color="Green" title="E53">A25</font></i>: 
<font color="Maroon" title="c10">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K6" title="GROEB_3:func.6">Low</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_3.html#E6:54"><i><font color="Green" title="E50">A6</font></i></a></i>;<br/><a NAME="E3:54_4_1_1_1"/>
<font color="Maroon" title="c10">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">s</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Olive" title="b2">t</font></span>)</span> where <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</font> is    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font> : ( <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c5">m</font> &amp; <font color="Olive" title="b2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K6" title="GROEB_3:func.6">Low</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="groeb_3.html#E1:54_4_1_1_1"><i><font color="Green" title="E52">A24</font></i></a>, <a class="txt" href="groeb_3.html#E4:54"><i><font color="Green" title="E48">A4</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">s</font>, <font color="Maroon" title="c12">t</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E5:54_4_1_1_1"/><i><font color="Green" title="E54">A26</font></i>: 
( <font color="Maroon" title="c10">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">s</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c12">t</font> &amp; <font color="Maroon" title="c11">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c5">m</font> &amp; <font color="Maroon" title="c12">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K6" title="GROEB_3:func.6">Low</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span> )
 ;<br/><a NAME="E6:54_4_1_1_1"/>
 <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c5">m</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="groeb_3.html#E5:54_4_1_1_1"><i><font color="Green" title="E54">A26</font></i></a>, <a class="ref" href="polynom7.html#T7" title="POLYNOM7:th.7">POLYNOM7:7</a></i>;<br/><a NAME="E7:54_4_1_1_1"/><b>then </b><i><font color="Green" title="E55">A27</font></i>: 
 <a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">s</font>
 <b>by </b><i><a class="txt" href="groeb_3.html#E5:54_4_1_1_1"><i><font color="Green" title="E54">A26</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <span class="p2">(<span class="default"><a href="groeb_3.html#K6" title="GROEB_3:func.6">Low</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c10">b</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <span class="p2">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="groeb_3.html#K6" title="GROEB_3:func.6">Low</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c12">t</font></span>)</span>
<b>by </b><i><a class="txt" href="groeb_3.html#E5:54_4_1_1_1"><i><font color="Green" title="E54">A26</font></i></a>, <a class="ref" href="polyred.html#T7" title="POLYRED:th.7">POLYRED:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <span class="p2">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c12">t</font></span>)</span>
<b>by </b><i><a class="txt" href="groeb_3.html#E5:54_4_1_1_1"><i><font color="Green" title="E54">A26</font></i></a>, <a class="txt" href="groeb_3.html#E1:54"><i><font color="Green" title="E45">A1</font></i></a>, <a class="ref" href="groeb_3.html#T31" target="_self" title="GROEB_3:th.31">Th31</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c10">b</font>
<b>by </b><i><a class="txt" href="groeb_3.html#E7:54_4_1_1_1"><i><font color="Green" title="E55">A27</font></i></a>, <a class="txt" href="groeb_3.html#E5:54_4_1_1_1"><i><font color="Green" title="E54">A26</font></i></a>, <a class="ref" href="polyred.html#T7" title="POLYRED:th.7">POLYRED:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="groeb_3.html#K6" title="GROEB_3:func.6">Low</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c10">b</font>
<b>by </b><i><a class="txt" href="groeb_3.html#E2:54_4_1_1_1"><i><font color="Green" title="E53">A25</font></i></a>, <a class="txt" href="groeb_3.html#E2:54"><i><font color="Green" title="E46">A2</font></i></a>, <a class="ref" href="groeb_3.html#T31" target="_self" title="GROEB_3:th.31">Th31</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_4_1_1_2"><b>case </b></a><a NAME="E1:54_4_1_1_2"/><i><font color="Green" title="E52">A28</font></i>: 
not <font color="Maroon" title="c10">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <span class="p2">(<span class="default"><a href="groeb_3.html#K6" title="GROEB_3:func.6">Low</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span></span>)</span>
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <span class="p2">(<span class="default"><a href="groeb_3.html#K6" title="GROEB_3:func.6">Low</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c10">b</font> = 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">L</font>
<b>by </b><i><a class="ref" href="polynom1.html#D9" title="POLYNOM1:def.9">POLYNOM1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="groeb_3.html#K6" title="GROEB_3:func.6">Low</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c10">b</font>
<b>by </b><i><a class="txt" href="groeb_3.html#E1:54_4_1_1_2"><i><font color="Green" title="E52">A28</font></i></a>, <a class="txt" href="groeb_3.html#E8:54"><i><font color="Green" title="E51">A23</font></i></a>, <a class="ref" href="polynom1.html#D9" title="POLYNOM1:def.9">POLYNOM1:def 9</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:54_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <span class="p2">(<span class="default"><a href="groeb_3.html#K6" title="GROEB_3:func.6">Low</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K6" title="GROEB_3:func.6">Low</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:54"/>
 <a href="groeb_3.html#K6" title="GROEB_3:func.6">Low</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K6" title="GROEB_3:func.6">Low</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_3.html#E3:54"><i><font color="Green" title="E47">A3</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>


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