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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">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_1.html#L3" title="VECTSP_1:struct.3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c3">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font>;<br/><b>let </b><font color="Maroon" title="c4">b</font> be   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>;<br/><b>let </b><font color="Maroon" title="c5">x</font> be   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>;<br/>









<b>set </b><font color="Maroon" title="c6">m</font> =  <a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_1_1"/>
( <font color="Maroon" title="c3">a</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="c2">L</font> or <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1"><b>case </b></a><a NAME="E1:31_1_1_1"/>
<font color="Maroon" title="c3">a</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="c2">L</font>
 ;<br/><div class="add"><a NAME="E2:31_1_1_1"/><b>then </b><i><font color="Green" title="E13">A1</font></i>: 
<font color="Maroon" title="c3">a</font> is <a href="rlvect_1.html#V8" title="RLVECT_1:attr.8">non-zero</a>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D13" title="RLVECT_1:def.13">RLVECT_1:def 13</a></i>;<br/><b>thus </b> <a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <span class="p1">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span>)</span>,<font color="Maroon" title="c5">x</font> = 
<span class="p1">(<span class="default"><a href="polynom7.html#K3" title="POLYNOM7:func.3">coefficient</a> <span class="p2">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <span class="p2">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <span class="p3">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom7.html#T12" target="_self" title="POLYNOM7:th.12">Th12</a></i>
<br/>.= 
<font color="Maroon" title="c3">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <span class="p2">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <span class="p3">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom7.html#T9" target="_self" title="POLYNOM7:th.9">Th9</a></i>
<br/>.= 
<font color="Maroon" title="c3">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">x</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom7.html#E2:31_1_1_1"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="polynom7.html#T10" target="_self" title="POLYNOM7:th.10">Th10</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_2"><b>case </b></a><a NAME="E1:31_1_1_2"/><i><font color="Green" title="E13">A2</font></i>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font>
 ;<br/><div class="add"><a NAME="E2:31_1_1_2"/>
for <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> holds  <span class="p1">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Olive" title="b1">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">b'</font> be   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_1_1_2_1_1_1"/>
( <font color="Maroon" title="c7">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> or <font color="Maroon" title="c7">b'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_2_1_1_1_1"><b>case </b></a><a NAME="E1:31_1_1_2_1_1_1_1"/><i><font color="Green" title="E14">A3</font></i>: 
<font color="Maroon" title="c7">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font>
 ;<br/><div class="add"><i><font color="Green" title="E15">A4</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K26" title="POLYNOM1:func.26">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font></span>)</span> = 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D24" title="POLYNOM1:def.24">POLYNOM1:def 24</a></i>
<br/>.= 
 <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="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>
;<br/><a NAME="E3:31_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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="polynom1.html#D14" title="POLYNOM1:def.14">POLYNOM1:def 14</a></i>;<br/><a NAME="E4:31_1_1_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E16">A5</font></i>: 
 <a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K26" title="POLYNOM1:func.26">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c3">a</font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom7.html#E2:31_1_1_2_1_1_1_1"><i><font color="Green" title="E15">A4</font></i></a>, <a class="ref" href="funct_7.html#D3" title="FUNCT_7:def.3">FUNCT_7:def 3</a></i>;<br/><a NAME="E5:31_1_1_2_1_1_1_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c3">a</font></span>)</span> <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"><font color="Maroon" title="c4">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/><a NAME="E6:31_1_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c3">a</font></span>)</span>
 <b>by </b><i><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"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c7">b'</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">b</font>
<b>by </b><i><a class="txt" href="polynom7.html#E1:31_1_1_2_1_1_1_1"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="polynom7.html#E4:31_1_1_2_1_1_1_1"><i><font color="Green" title="E16">A5</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font>
<b>by </b><i><a class="txt" href="polynom7.html#E1:31_1_1_2"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="funcop_1.html#T87" title="FUNCOP_1:th.87">FUNCOP_1:87</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_2_1_1_1_2"><b>case </b></a><a NAME="E1:31_1_1_2_1_1_1_2"/>
<font color="Maroon" title="c7">b'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font>
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c7">b'</font> = 
<span class="p1">(<span class="default"><a href="polynom1.html#K26" title="POLYNOM1:func.26">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c7">b'</font>
<b>by </b><i><a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font>
<b>by </b><i><a class="ref" href="polynom1.html#T81" title="POLYNOM1:th.81">POLYNOM1:81</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:31_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c7">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E3:31_1_1_2"/><b>then </b><i><font color="Green" title="E14">A6</font></i>: 
<span class="p1">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span>)</span> <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> <span class="p2">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font>
 ;<br/><b>thus </b> <a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <span class="p1">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span>)</span>,<font color="Maroon" title="c5">x</font> = 
<span class="p1">(<span class="default"><a href="polynom7.html#K3" title="POLYNOM7:func.3">coefficient</a> <span class="p2">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <span class="p2">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <span class="p3">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom7.html#T12" target="_self" title="POLYNOM7:th.12">Th12</a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font>
<b>by </b><i><a class="txt" href="polynom7.html#E3:31_1_1_2"><i><font color="Green" title="E14">A6</font></i></a>, <a class="ref" href="vectsp_1.html#T39" title="VECTSP_1:th.39">VECTSP_1:39</a></i>
<br/>.= 
<font color="Maroon" title="c3">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">x</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom7.html#E1:31_1_1_2"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="vectsp_1.html#T39" title="VECTSP_1:th.39">VECTSP_1:39</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:31"/>
 <a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <span class="p1">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span>)</span>,<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">x</font></span>)</span>
 ;<br/>


</div>
