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<b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="vectsp_1.html#V7">distributive</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">b</font> be   <a href="polynom1.html#NM1">bag</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">f</font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">L</font></span>)</span> <a href="finseq_2.html#K3">*</a> ;<br/><b>let </b><font color="Maroon">s</font> be   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">L</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:87"/><i><font color="Green">E42</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span>
 <b>and </b><br/><a NAME="E2:87"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="uproots.html#K1" target="_self">canFS</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span>
 <b>and </b><br/><a NAME="E3:87"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="uproots.html#K9" target="_self">fpoly_mult_root</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E4:87"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:87"/><b>then </b><i><font color="Green">E45</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">s</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Cf</font> =  <a href="polynom1.html#K5">Card</a> <font color="Maroon">f</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E7:87"/><i><font color="Green">E46</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="wsierp_1.html#K7">Sum</a> <font color="Maroon">Cf</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#T30">POLYNOM1:30</a></i>;<br/>
<b>consider </b><font color="Maroon">g</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E9:87"/><i><font color="Green">E47</font></i>: 
 <a href="uproots.html#K4" target="_self">degree</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="wsierp_1.html#K7">Sum</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E10:87"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 <b>by </b><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">E51</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">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">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:87_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font>
 ;<br/>

<a NAME="E2:87_1_1"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <font color="Maroon">b</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:87_1_1"/>
 <a href="polynom1.html#K11">support</a> <font color="Maroon">b</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#T41">POLYNOM1:41</a></i>;<br/>
<b>hence </b><a NAME="E4:87_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>thus </b> <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5">Card</a> <font color="Maroon">f</font></span>)</span> = 
 <a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="card_3.html#D2">CARD_3:def 2</a></i>
<br/>.= 
 <a href="finsop_1.html#K5">dom</a> <font color="Maroon">g</font>
<b>by </b><i>, , <a class="ref" href="uproots.html#T3" target="_self">Th3</a>, , <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>
;<br/><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E3:87_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5">Card</a> <font color="Maroon">f</font></span>)</span>
 ;<br/><a NAME="E4:87_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="card_3.html#D2">CARD_3:def 2</a></i>;<br/><a NAME="E5:87_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="uproots.html#K9" target="_self">fpoly_mult_root</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E6:87_1"/><b>then </b><i><font color="Green">E55</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E7:87_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">g</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="uproots.html#T3" target="_self">Th3</a>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><b>thus </b><font color="Maroon">Cf</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i</font> = 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, <a class="ref" href="card_3.html#D2">CARD_3:def 2</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i</font>
<b>by </b><i>, , , , <a class="ref" href="finseq_4.html#D4">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">len</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="uproots.html#K4" target="_self">degree</a> <font color="Maroon">b</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T17">FINSEQ_1:17</a></i>;<br/>


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