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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V3">Abelian</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="group_1.html#V7">commutative</a> <a href="vectsp_1.html#V6">right_unital</a> <a href="vectsp_1.html#V7">distributive</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="polynom1.html#K16">EmptyBag</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being    <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">c<sub>1</sub></font></span>)</span> <a href="finseq_2.html#K3">*</a> , <font color="Maroon">c<sub>4</sub></font> being   <a href="struct_0.html#NM8">FinSequence</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E2:85"/><i><font color="Green">E69</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></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> <span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E3:85"/>
<font color="Maroon">c<sub>4</sub></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> <span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E4:85"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></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">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E5:85"/><i><font color="Green">E70</font></i>: 
 <a href="uproots.html#K10" target="_self">poly_with_roots</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_4.html#K3">Product</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="uproots.html#D11" target="_self">Def11</a></i>;<br/>
<a NAME="E6:85"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11">support</a> <span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="bagorder.html#T19">BAGORDER:19</a>, <a class="ref" href="card_1.html#T78">CARD_1:78</a></i>;<br/>
<a NAME="E7:85"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_2.html#K3">*</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="uproots.html#E2:85"><i><font color="Green">E69</font></i></a>, <a class="ref" href="finseq_1.html#T32">FINSEQ_1:32</a></i>;<br/>
<a NAME="E8:85"/><b>then </b>
 <a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> 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">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="dtconstr.html#T20">DTCONSTR:20</a></i>;<br/>
<b>then </b> <a href="group_4.html#K3">Product</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> = 
 <a href="group_1.html#K2">1.</a> <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="fvsum_1.html#T98">FVSUM_1:98</a></i>
<br/>.= 
 <a href="polynom3.html#K13">1_.</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="polynom3.html#T37">POLYNOM3:37</a></i>
;<br/>
<b>hence </b><a NAME="E10:85"/>
 <a href="uproots.html#K10" target="_self">poly_with_roots</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="algseq_1.html#K5">%&gt;</a></span>
 <b>by </b><i><a class="txt" href="uproots.html#E5:85"><i><font color="Green">E70</font></i></a>, <a class="ref" href="uproots.html#T33" target="_self">Th33</a></i>;<br/>


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