<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</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="vectsp_1.html#V7" title="VECTSP_1:attr.7">distributive</a>  <a href="vectsp_1.html#L3" title="VECTSP_1:struct.3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">b</font> be   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>;<br/><b>let </b><font color="Maroon" title="c3">f</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16" title="POLYNOM3:func.16">Polynom-Ring</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> ;<br/><b>let </b><font color="Maroon" title="c4">s</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">L</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:87"/><i><font color="Green" title="E71">A1</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c2">b</font></span>)</span>
 <b>and </b><br/><a NAME="E2:87"/><i><font color="Green" title="E72">A2</font></i>: 
<font color="Maroon" title="c4">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c2">b</font></span>)</span>
 <b>and </b><br/><a NAME="E3:87"/><i><font color="Green" title="E73">A3</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c3">f</font> holds <br/><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="uproots.html#K9" title="UPROOTS:func.9">fpoly_mult_root</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E4:87"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c2">b</font></span>)</span>
 
<b>by </b><i><a class="txt" href="uproots.html#E2:87"><i><font color="Green" title="E72">A2</font></i></a>, <a class="ref" href="uproots.html#D1" target="_self" title="UPROOTS:def.1">Def1</a></i>;<br/>
<a NAME="E5:87"/><b>then </b><i><font color="Green" title="E74">A4</font></i>: 
 <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c4">s</font>
 
<b>by </b><i><a class="txt" href="uproots.html#E1:87"><i><font color="Green" title="E71">A1</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">Cf</font> =  <a href="polynom1.html#K5" title="POLYNOM1:func.5">Card</a> <font color="Maroon" title="c3">f</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E7:87"/><i><font color="Green" title="E75">A5</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Sum</a> <font color="Maroon" title="c5">Cf</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#T30" title="POLYNOM1:th.30">POLYNOM1:30</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">g</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E9:87"/><i><font color="Green" title="E76">A6</font></i>: 
 <a href="uproots.html#K4" title="UPROOTS:func.4">degree</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Sum</a> <font color="Maroon" title="c6">g</font>
 <b>and </b><br/><a NAME="E10:87"/><i><font color="Green" title="E77">A7</font></i>: 
<font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="uproots.html#D4" target="_self" title="UPROOTS:def.4">Def4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1"><b>now </b></a><div class="add"><a NAME="E1:87_1"/><i><font color="Green" title="E78">A8</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">b</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:87_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">s</font>
 ;<br/>

<a NAME="E2:87_1_1"/><b>then </b><i><font color="Green" title="E79">A9</font></i>: 
<font color="Maroon" title="c7">x</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="c2">b</font>
 
<b>by </b><i><a class="txt" href="uproots.html#E2:87"><i><font color="Green" title="E72">A2</font></i></a>, <a class="ref" href="uproots.html#T5" target="_self" title="UPROOTS:th.5">Th5</a></i>;<br/>
<a NAME="E3:87_1_1"/>
 <a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c2">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">b</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#T41" title="POLYNOM1:th.41">POLYNOM1:41</a></i>;<br/>
<b>hence </b><a NAME="E4:87_1_1"/>
<font color="Maroon" title="c7">x</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> <font color="Maroon" title="c2">b</font>
 <b>by </b><i><a class="txt" href="uproots.html#E2:87_1_1"><i><font color="Green" title="E79">A9</font></i></a></i>;<br/>


</div><b>end;</b></div><b>thus </b> <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5" title="POLYNOM1:func.5">Card</a> <font color="Maroon" title="c3">f</font></span>)</span> = 
 <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c3">f</font>
<b>by </b><i><a class="ref" href="card_3.html#D2" title="CARD_3:def.2">CARD_3:def 2</a></i>
<br/>.= 
 <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c6">g</font>
<b>by </b><i><a class="txt" href="uproots.html#E2:87"><i><font color="Green" title="E72">A2</font></i></a>, <a class="txt" href="uproots.html#E5:87"><i><font color="Green" title="E74">A4</font></i></a>, <a class="txt" href="uproots.html#E10:87"><i><font color="Green" title="E77">A7</font></i></a>, <a class="txt" href="uproots.html#E1:87_1"><i><font color="Green" title="E78">A8</font></i></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>
;<br/><b>let </b><font color="Maroon" title="c7">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E3:87_1"/><i><font color="Green" title="E79">A10</font></i>: 
<font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5" title="POLYNOM1:func.5">Card</a> <font color="Maroon" title="c3">f</font></span>)</span>
 ;<br/><a NAME="E4:87_1"/><i><font color="Green" title="E80">A11</font></i>: 
<font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="txt" href="uproots.html#E3:87_1"><i><font color="Green" title="E79">A10</font></i></a>, <a class="ref" href="card_3.html#D2" title="CARD_3:def.2">CARD_3:def 2</a></i>;<br/><a NAME="E5:87_1"/><b>then </b>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="uproots.html#K9" title="UPROOTS:func.9">fpoly_mult_root</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c7">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c7">i</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="uproots.html#E3:87"><i><font color="Green" title="E73">A3</font></i></a></i>;<br/><a NAME="E6:87_1"/><b>then </b><i><font color="Green" title="E81">A12</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c7">i</font></span>)</span>
 <b>by </b><i><a class="ref" href="uproots.html#D10" target="_self" title="UPROOTS:def.10">Def10</a></i>;<br/><a NAME="E7:87_1"/><i><font color="Green" title="E82">A13</font></i>: 
<font color="Maroon" title="c6">g</font> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="uproots.html#E2:87"><i><font color="Green" title="E72">A2</font></i></a>, <a class="txt" href="uproots.html#E5:87"><i><font color="Green" title="E74">A4</font></i></a>, <a class="txt" href="uproots.html#E10:87"><i><font color="Green" title="E77">A7</font></i></a>, <a class="txt" href="uproots.html#E4:87_1"><i><font color="Green" title="E80">A11</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><b>thus </b><font color="Maroon" title="c5">Cf</font> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">.</a> <font color="Maroon" title="c7">i</font> = 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span>
<b>by </b><i><a class="txt" href="uproots.html#E4:87_1"><i><font color="Green" title="E80">A11</font></i></a>, <a class="ref" href="card_3.html#D2" title="CARD_3:def.2">CARD_3:def 2</a></i>
<br/>.= 
<font color="Maroon" title="c6">g</font> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">.</a> <font color="Maroon" title="c7">i</font>
<b>by </b><i><a class="txt" href="uproots.html#E5:87"><i><font color="Green" title="E74">A4</font></i></a>, <a class="txt" href="uproots.html#E4:87_1"><i><font color="Green" title="E80">A11</font></i></a>, <a class="txt" href="uproots.html#E6:87_1"><i><font color="Green" title="E81">A12</font></i></a>, <a class="txt" href="uproots.html#E7:87_1"><i><font color="Green" title="E82">A13</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:87"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="uproots.html#K4" title="UPROOTS:func.4">degree</a> <font color="Maroon" title="c2">b</font>
 <b>by </b><i><a class="txt" href="uproots.html#E7:87"><i><font color="Green" title="E75">A5</font></i></a>, <a class="txt" href="uproots.html#E9:87"><i><font color="Green" title="E76">A6</font></i></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></i>;<br/>


</div>
