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

<span class="kw">let </span><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="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds   <a href="uproots.html#K10" title="UPROOTS:func.10">poly_with_roots</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">c</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> </span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Olive" title="b1">c</font></span>)</span>,<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="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="uproots.html#K10" title="UPROOTS:func.10">poly_with_roots</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">c</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> </span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c2">c</font></span>)</span>,<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="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span></span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">b</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">c</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">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#K14" title="POLYNOM3:func.14">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="c5">s</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E2:89"/><span class="lab"><font color="Green" title="E67">A1</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">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="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">c</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> </span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:89"/><span class="lab"><font color="Green" title="E68">A2</font></span>: 
<font color="Maroon" title="c5">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="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">c</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> </span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:89"/><span class="lab"><font color="Green" title="E69">A3</font></span>: 
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="c4">f</font> holds <br/><font color="Maroon" title="c4">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="c5">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"><span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">c</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:89"/><span class="lab"><font color="Green" title="E70">A4</font></span>: 
 <a href="uproots.html#K10" title="UPROOTS:func.10">poly_with_roots</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">c</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> </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="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="uproots.html#D11" target="_self" title="UPROOTS:def.11">Def11</a></span>;<br/>
<a NAME="E6:89"/><span class="lab"><font color="Green" title="E71">A5</font></span>: 
 <a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">c</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">c</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="uproots.html#T10" target="_self" title="UPROOTS:th.10">Th10</a></span>;<br/>
<a NAME="E7:89"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A6</font></span>: 
<font color="Maroon" title="c5">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c2">c</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:89"><span class="lab"><font color="Green" title="E68">A2</font></span></a>, <a class="ref" href="uproots.html#T6" target="_self" title="UPROOTS:th.6">Th6</a></span>;<br/>
<a NAME="E8:89"/><span class="lab"><font color="Green" title="E73">A7</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">c</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:89"><span class="lab"><font color="Green" title="E71">A5</font></span></a>, <a class="ref" href="card_1.html#T50" title="CARD_1:th.50">CARD_1:50</a></span>;<br/>
<a NAME="E9:89"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A8</font></span>: 
1 <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="c4">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:89"><span class="lab"><font color="Green" title="E67">A1</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E10:89"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A9</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/>
<a NAME="E11:89"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:89"><span class="lab"><font color="Green" title="E68">A2</font></span></a>, <a class="txt" href="#E8:89"><span class="lab"><font color="Green" title="E73">A7</font></span></a>, <a class="ref" href="uproots.html#T5" target="_self" title="UPROOTS:th.5">Th5</a></span>;<br/>
<a NAME="E12:89"/><span class="kw">then </span>
1 <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="c5">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E76">A10</font></span>: <font color="Maroon" title="c5">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<font color="Maroon" title="c5">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>
<br/>.= 
<font color="Maroon" title="c2">c</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:89"><span class="lab"><font color="Green" title="E72">A6</font></span></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></span>
;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">f1</font> =  <a href="uproots.html#K9" title="UPROOTS:func.9">fpoly_mult_root</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">c</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span>;<br/>
<a NAME="E14:89"/>
<font color="Maroon" title="c2">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">c</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E15:89"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">c</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:89"><span class="lab"><font color="Green" title="E76">A10</font></span></a>, <a class="ref" href="uproots.html#T9" target="_self" title="UPROOTS:th.9">Th9</a></span>;<br/>
<a NAME="E16:89"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A11</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="uproots.html#K9" title="UPROOTS:func.9">fpoly_mult_root</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">c</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="uproots.html#D10" target="_self" title="UPROOTS:def.10">Def10</a></span>;<br/>
<a NAME="E17:89"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A12</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="uproots.html#K9" title="UPROOTS:func.9">fpoly_mult_root</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">c</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E18:89"/>
<font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> 1</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:89"><span class="lab"><font color="Green" title="E67">A1</font></span></a>, <a class="txt" href="#E6:89"><span class="lab"><font color="Green" title="E71">A5</font></span></a>, <a class="ref" href="card_1.html#T50" title="CARD_1:th.50">CARD_1:50</a>, <a class="ref" href="finseq_5.html#T15" title="FINSEQ_5:th.15">FINSEQ_5:15</a></span>;<br/>
<a NAME="E19:89"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A13</font></span>: 
 <a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:89"><span class="lab"><font color="Green" title="E75">A9</font></span></a>, <a class="ref" href="pre_poly.html#T1" title="PRE_POLY:th.1">PRE_POLY:1</a></span>;<br/>
<a NAME="E20:89"/><span class="lab"><font color="Green" title="E80">A14</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <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="c5">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">c</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:89"><span class="lab"><font color="Green" title="E69">A3</font></span></a>, <a class="txt" href="#E9:89"><span class="lab"><font color="Green" title="E74">A8</font></span></a></span>;<br/>
<a NAME="E21:89"/>
 <a href="uproots.html#K9" title="UPROOTS:func.9">fpoly_mult_root</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">c</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K9" title="UPROOTS:func.9">fpoly_mult_root</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">c</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:89"><span class="lab"><font color="Green" title="E77">A11</font></span></a>, <a class="ref" href="finseq_5.html#T15" title="FINSEQ_5:th.15">FINSEQ_5:15</a></span>;<br/>
<span class="kw">hence </span> <a href="uproots.html#K10" title="UPROOTS:func.10">poly_with_roots</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">c</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> </span>)</span> = 
<span class="p1">(<span class="default"><a href="uproots.html#K9" title="UPROOTS:func.9">fpoly_mult_root</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">c</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:89"><span class="lab"><font color="Green" title="E70">A4</font></span></a>, <a class="txt" href="#E19:89"><span class="lab"><font color="Green" title="E79">A13</font></span></a>, <a class="txt" href="#E20:89"><span class="lab"><font color="Green" title="E80">A14</font></span></a>, <a class="ref" href="finsop_1.html#T12" title="FINSOP_1:th.12">FINSOP_1:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="uproots.html#K9" title="UPROOTS:func.9">fpoly_mult_root</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">c</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:89"><span class="lab"><font color="Green" title="E78">A12</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>
<br/>.= 
<span class="p1"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c2">c</font></span>)</span>,<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="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:89"><span class="lab"><font color="Green" title="E76">A10</font></span></a>, <a class="txt" href="#E17:89"><span class="lab"><font color="Green" title="E78">A12</font></span></a>, <a class="ref" href="uproots.html#D10" target="_self" title="UPROOTS:def.10">Def10</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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