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<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a> <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a> <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="uproots.html#K10" title="UPROOTS:func.10">poly_with_roots</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">EmptyBag</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1"><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">%&gt;</a></span></span><br/>

<a NAME="E1:86"/><i><font color="Green" title="E67">A1</font></i>: 
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <span class="p1">(<span class="default"><a href="polynom3.html#K13" title="POLYNOM3:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p1">(<span class="default"><a href="polynom3.html#K13" title="POLYNOM3:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">L</font></span>)</span>
 
;<br/>
<b>set </b><font color="Maroon" title="c2">b</font> =  <a href="polynom1.html#K14" title="POLYNOM1:func.14">EmptyBag</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>;<br/>
<b>consider </b><font color="Maroon" title="c3">f</font> being    <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#K13" title="POLYNOM3:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> , <font color="Maroon" title="c4">s</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E3:86"/><i><font color="Green" title="E68">A2</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#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <span class="p2">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">EmptyBag</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E4:86"/>
<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#K9" title="POLYNOM1:func.9">support</a> <span class="p2">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">EmptyBag</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E5:86"/>
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="finseq_1.html#K4" title="FINSEQ_1:func.4">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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">EmptyBag</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E6:86"/><i><font color="Green" title="E69">A3</font></i>: 
 <a href="uproots.html#K10" title="UPROOTS:func.10">poly_with_roots</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">EmptyBag</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</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>
 <b>by </b><i><a class="ref" href="uproots.html#D11" target="_self" title="UPROOTS:def.11">Def11</a></i>;<br/>
<a NAME="E7:86"/>
<font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="polynom3.html#K13" title="POLYNOM3:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:86"><i><font color="Green" title="E68">A2</font></i></a>, <a class="ref" href="bagorder.html#T19" title="BAGORDER:th.19">BAGORDER:19</a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a></i>;<br/>
<a NAME="E8:86"/><b>then </b>
 <a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> 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#K13" title="POLYNOM3:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">L</font></span>)</span>
 
;<br/>
<b>then </b> <a href="group_4.html#K3" title="GROUP_4:func.3">Product</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="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p1">(<span class="default"><a href="polynom3.html#K13" title="POLYNOM3:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">L</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:86"><i><font color="Green" title="E67">A1</font></i></a>, <a class="ref" href="group_4.html#T11" title="GROUP_4:th.11">GROUP_4:11</a></i>
<br/>.= 
 <a href="polynom3.html#K10" title="POLYNOM3:func.10">1_.</a> <font color="Maroon" title="c1">L</font>
<b>by </b><i><a class="ref" href="polynom3.html#T37" title="POLYNOM3:th.37">POLYNOM3:37</a></i>
;<br/>
<b>hence </b><a NAME="E10:86"/>
 <a href="uproots.html#K10" title="UPROOTS:func.10">poly_with_roots</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">EmptyBag</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1"><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">%&gt;</a></span>
 <b>by </b><i><a class="txt" href="#E6:86"><i><font color="Green" title="E69">A3</font></i></a>, <a class="ref" href="uproots.html#T33" target="_self" title="UPROOTS:th.33">Th33</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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